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Theorem nqereu 10344
Description: There is a unique element of Q equivalent to each element of N × N. (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
nqereu (𝐴 ∈ (N × N) → ∃!𝑥Q 𝑥 ~Q 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem nqereu
Dummy variables 𝑎 𝑏 𝑦 𝑐 𝑑 𝑚 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp2 5547 . . 3 (𝐴 ∈ (N × N) ↔ ∃𝑎N𝑏N 𝐴 = ⟨𝑎, 𝑏⟩)
2 pion 10294 . . . . . . . . 9 (𝑏N𝑏 ∈ On)
3 suceloni 7512 . . . . . . . . 9 (𝑏 ∈ On → suc 𝑏 ∈ On)
42, 3syl 17 . . . . . . . 8 (𝑏N → suc 𝑏 ∈ On)
5 vex 3447 . . . . . . . . 9 𝑏 ∈ V
65sucid 6242 . . . . . . . 8 𝑏 ∈ suc 𝑏
7 eleq2 2881 . . . . . . . . 9 (𝑦 = suc 𝑏 → (𝑏𝑦𝑏 ∈ suc 𝑏))
87rspcev 3574 . . . . . . . 8 ((suc 𝑏 ∈ On ∧ 𝑏 ∈ suc 𝑏) → ∃𝑦 ∈ On 𝑏𝑦)
94, 6, 8sylancl 589 . . . . . . 7 (𝑏N → ∃𝑦 ∈ On 𝑏𝑦)
109adantl 485 . . . . . 6 ((𝑎N𝑏N) → ∃𝑦 ∈ On 𝑏𝑦)
11 elequ2 2127 . . . . . . . . . . . . . 14 (𝑦 = 𝑚 → (𝑏𝑦𝑏𝑚))
1211imbi1d 345 . . . . . . . . . . . . 13 (𝑦 = 𝑚 → ((𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩) ↔ (𝑏𝑚 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
13122ralbidv 3167 . . . . . . . . . . . 12 (𝑦 = 𝑚 → (∀𝑎N𝑏N (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩) ↔ ∀𝑎N𝑏N (𝑏𝑚 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
14 opeq1 4766 . . . . . . . . . . . . . . . . . . 19 (𝑐 = 𝑎 → ⟨𝑐, 𝑑⟩ = ⟨𝑎, 𝑑⟩)
1514breq2d 5045 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝑎 → (𝑥 ~Q𝑐, 𝑑⟩ ↔ 𝑥 ~Q𝑎, 𝑑⟩))
1615rexbidv 3259 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑎 → (∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩ ↔ ∃𝑥Q 𝑥 ~Q𝑎, 𝑑⟩))
1716imbi2d 344 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑎 → ((𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ↔ (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑑⟩)))
18 elequ1 2119 . . . . . . . . . . . . . . . . 17 (𝑑 = 𝑏 → (𝑑𝑚𝑏𝑚))
19 opeq2 4768 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑏 → ⟨𝑎, 𝑑⟩ = ⟨𝑎, 𝑏⟩)
2019breq2d 5045 . . . . . . . . . . . . . . . . . 18 (𝑑 = 𝑏 → (𝑥 ~Q𝑎, 𝑑⟩ ↔ 𝑥 ~Q𝑎, 𝑏⟩))
2120rexbidv 3259 . . . . . . . . . . . . . . . . 17 (𝑑 = 𝑏 → (∃𝑥Q 𝑥 ~Q𝑎, 𝑑⟩ ↔ ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
2218, 21imbi12d 348 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑏 → ((𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑑⟩) ↔ (𝑏𝑚 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
2317, 22cbvral2vw 3411 . . . . . . . . . . . . . . 15 (∀𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ↔ ∀𝑎N𝑏N (𝑏𝑚 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
2423ralbii 3136 . . . . . . . . . . . . . 14 (∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ↔ ∀𝑚𝑦𝑎N𝑏N (𝑏𝑚 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
25 rexnal 3204 . . . . . . . . . . . . . . . . . . 19 (∃𝑧 ∈ (N × N) ¬ (⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏) ↔ ¬ ∀𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏))
26 pm4.63 401 . . . . . . . . . . . . . . . . . . . . 21 (¬ (⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏) ↔ (⟨𝑎, 𝑏⟩ ~Q 𝑧 ∧ (2nd𝑧) <N 𝑏))
27 xp2nd 7708 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ (N × N) → (2nd𝑧) ∈ N)
28 ltpiord 10302 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((2nd𝑧) ∈ N𝑏N) → ((2nd𝑧) <N 𝑏 ↔ (2nd𝑧) ∈ 𝑏))
2928ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏N ∧ (2nd𝑧) ∈ N) → ((2nd𝑧) <N 𝑏 ↔ (2nd𝑧) ∈ 𝑏))
3027, 29sylan2 595 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏N𝑧 ∈ (N × N)) → ((2nd𝑧) <N 𝑏 ↔ (2nd𝑧) ∈ 𝑏))
3130adantll 713 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑎N𝑏N) ∧ 𝑧 ∈ (N × N)) → ((2nd𝑧) <N 𝑏 ↔ (2nd𝑧) ∈ 𝑏))
3231anbi2d 631 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎N𝑏N) ∧ 𝑧 ∈ (N × N)) → ((⟨𝑎, 𝑏⟩ ~Q 𝑧 ∧ (2nd𝑧) <N 𝑏) ↔ (⟨𝑎, 𝑏⟩ ~Q 𝑧 ∧ (2nd𝑧) ∈ 𝑏)))
3326, 32syl5bb 286 . . . . . . . . . . . . . . . . . . . 20 (((𝑎N𝑏N) ∧ 𝑧 ∈ (N × N)) → (¬ (⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏) ↔ (⟨𝑎, 𝑏⟩ ~Q 𝑧 ∧ (2nd𝑧) ∈ 𝑏)))
3433rexbidva 3258 . . . . . . . . . . . . . . . . . . 19 ((𝑎N𝑏N) → (∃𝑧 ∈ (N × N) ¬ (⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏) ↔ ∃𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 ∧ (2nd𝑧) ∈ 𝑏)))
3525, 34bitr3id 288 . . . . . . . . . . . . . . . . . 18 ((𝑎N𝑏N) → (¬ ∀𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏) ↔ ∃𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 ∧ (2nd𝑧) ∈ 𝑏)))
36 xp1st 7707 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ (N × N) → (1st𝑧) ∈ N)
37 elequ2 2127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 = 𝑏 → (𝑑𝑚𝑑𝑏))
3837imbi1d 345 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 = 𝑏 → ((𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ↔ (𝑑𝑏 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩)))
39382ralbidv 3167 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 = 𝑏 → (∀𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ↔ ∀𝑐N𝑑N (𝑑𝑏 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩)))
4039rspccv 3571 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) → (𝑏𝑦 → ∀𝑐N𝑑N (𝑑𝑏 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩)))
41 opeq1 4766 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑐 = (1st𝑧) → ⟨𝑐, 𝑑⟩ = ⟨(1st𝑧), 𝑑⟩)
4241breq2d 5045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = (1st𝑧) → (𝑥 ~Q𝑐, 𝑑⟩ ↔ 𝑥 ~Q ⟨(1st𝑧), 𝑑⟩))
4342rexbidv 3259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = (1st𝑧) → (∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩ ↔ ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), 𝑑⟩))
4443imbi2d 344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 = (1st𝑧) → ((𝑑𝑏 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ↔ (𝑑𝑏 → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), 𝑑⟩)))
4544ralbidv 3165 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑐 = (1st𝑧) → (∀𝑑N (𝑑𝑏 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ↔ ∀𝑑N (𝑑𝑏 → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), 𝑑⟩)))
4645rspccv 3571 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∀𝑐N𝑑N (𝑑𝑏 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) → ((1st𝑧) ∈ N → ∀𝑑N (𝑑𝑏 → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), 𝑑⟩)))
47 eleq1 2880 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑑 = (2nd𝑧) → (𝑑𝑏 ↔ (2nd𝑧) ∈ 𝑏))
48 opeq2 4768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑑 = (2nd𝑧) → ⟨(1st𝑧), 𝑑⟩ = ⟨(1st𝑧), (2nd𝑧)⟩)
4948breq2d 5045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑑 = (2nd𝑧) → (𝑥 ~Q ⟨(1st𝑧), 𝑑⟩ ↔ 𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩))
5049rexbidv 3259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑑 = (2nd𝑧) → (∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), 𝑑⟩ ↔ ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩))
5147, 50imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑑 = (2nd𝑧) → ((𝑑𝑏 → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), 𝑑⟩) ↔ ((2nd𝑧) ∈ 𝑏 → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩)))
5251rspccv 3571 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∀𝑑N (𝑑𝑏 → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), 𝑑⟩) → ((2nd𝑧) ∈ N → ((2nd𝑧) ∈ 𝑏 → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩)))
5346, 52syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑐N𝑑N (𝑑𝑏 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) → ((1st𝑧) ∈ N → ((2nd𝑧) ∈ N → ((2nd𝑧) ∈ 𝑏 → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩))))
5440, 53syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) → (𝑏𝑦 → ((1st𝑧) ∈ N → ((2nd𝑧) ∈ N → ((2nd𝑧) ∈ 𝑏 → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩)))))
5554imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ∧ 𝑏𝑦) → ((1st𝑧) ∈ N → ((2nd𝑧) ∈ N → ((2nd𝑧) ∈ 𝑏 → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩))))
5636, 55syl5 34 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ∧ 𝑏𝑦) → (𝑧 ∈ (N × N) → ((2nd𝑧) ∈ N → ((2nd𝑧) ∈ 𝑏 → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩))))
5727, 56mpdi 45 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ∧ 𝑏𝑦) → (𝑧 ∈ (N × N) → ((2nd𝑧) ∈ 𝑏 → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩)))
58573imp 1108 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ∧ 𝑏𝑦) ∧ 𝑧 ∈ (N × N) ∧ (2nd𝑧) ∈ 𝑏) → ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩)
59 1st2nd2 7714 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ (N × N) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
6059breq2d 5045 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ (N × N) → (𝑥 ~Q 𝑧𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩))
6160rexbidv 3259 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ (N × N) → (∃𝑥Q 𝑥 ~Q 𝑧 ↔ ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩))
62613ad2ant2 1131 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ∧ 𝑏𝑦) ∧ 𝑧 ∈ (N × N) ∧ (2nd𝑧) ∈ 𝑏) → (∃𝑥Q 𝑥 ~Q 𝑧 ↔ ∃𝑥Q 𝑥 ~Q ⟨(1st𝑧), (2nd𝑧)⟩))
6358, 62mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . 24 (((∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ∧ 𝑏𝑦) ∧ 𝑧 ∈ (N × N) ∧ (2nd𝑧) ∈ 𝑏) → ∃𝑥Q 𝑥 ~Q 𝑧)
64 enqer 10336 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ~Q Er (N × N)
6564a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑎, 𝑏⟩ ~Q 𝑧𝑥 ~Q 𝑧) → ~Q Er (N × N))
66 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑎, 𝑏⟩ ~Q 𝑧𝑥 ~Q 𝑧) → 𝑥 ~Q 𝑧)
67 simpl 486 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑎, 𝑏⟩ ~Q 𝑧𝑥 ~Q 𝑧) → ⟨𝑎, 𝑏⟩ ~Q 𝑧)
6865, 66, 67ertr4d 8295 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((⟨𝑎, 𝑏⟩ ~Q 𝑧𝑥 ~Q 𝑧) → 𝑥 ~Q𝑎, 𝑏⟩)
6968ex 416 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨𝑎, 𝑏⟩ ~Q 𝑧 → (𝑥 ~Q 𝑧𝑥 ~Q𝑎, 𝑏⟩))
7069reximdv 3235 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑎, 𝑏⟩ ~Q 𝑧 → (∃𝑥Q 𝑥 ~Q 𝑧 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
7163, 70syl5com 31 . . . . . . . . . . . . . . . . . . . . . . 23 (((∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ∧ 𝑏𝑦) ∧ 𝑧 ∈ (N × N) ∧ (2nd𝑧) ∈ 𝑏) → (⟨𝑎, 𝑏⟩ ~Q 𝑧 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
72713expia 1118 . . . . . . . . . . . . . . . . . . . . . 22 (((∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ∧ 𝑏𝑦) ∧ 𝑧 ∈ (N × N)) → ((2nd𝑧) ∈ 𝑏 → (⟨𝑎, 𝑏⟩ ~Q 𝑧 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
7372impcomd 415 . . . . . . . . . . . . . . . . . . . . 21 (((∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ∧ 𝑏𝑦) ∧ 𝑧 ∈ (N × N)) → ((⟨𝑎, 𝑏⟩ ~Q 𝑧 ∧ (2nd𝑧) ∈ 𝑏) → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
7473rexlimdva 3246 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) ∧ 𝑏𝑦) → (∃𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 ∧ (2nd𝑧) ∈ 𝑏) → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
7574ex 416 . . . . . . . . . . . . . . . . . . 19 (∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) → (𝑏𝑦 → (∃𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 ∧ (2nd𝑧) ∈ 𝑏) → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
7675com3r 87 . . . . . . . . . . . . . . . . . 18 (∃𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 ∧ (2nd𝑧) ∈ 𝑏) → (∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) → (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
7735, 76syl6bi 256 . . . . . . . . . . . . . . . . 17 ((𝑎N𝑏N) → (¬ ∀𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏) → (∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) → (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))))
7877com13 88 . . . . . . . . . . . . . . . 16 (∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) → (¬ ∀𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏) → ((𝑎N𝑏N) → (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))))
79 mulcompi 10311 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ·N 𝑏) = (𝑏 ·N 𝑎)
80 enqbreq 10334 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎N𝑏N) ∧ (𝑎N𝑏N)) → (⟨𝑎, 𝑏⟩ ~Q𝑎, 𝑏⟩ ↔ (𝑎 ·N 𝑏) = (𝑏 ·N 𝑎)))
8180anidms 570 . . . . . . . . . . . . . . . . . . . 20 ((𝑎N𝑏N) → (⟨𝑎, 𝑏⟩ ~Q𝑎, 𝑏⟩ ↔ (𝑎 ·N 𝑏) = (𝑏 ·N 𝑎)))
8279, 81mpbiri 261 . . . . . . . . . . . . . . . . . . 19 ((𝑎N𝑏N) → ⟨𝑎, 𝑏⟩ ~Q𝑎, 𝑏⟩)
83 opelxpi 5560 . . . . . . . . . . . . . . . . . . . 20 ((𝑎N𝑏N) → ⟨𝑎, 𝑏⟩ ∈ (N × N))
84 breq1 5036 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 ~Q 𝑧 ↔ ⟨𝑎, 𝑏⟩ ~Q 𝑧))
85 vex 3447 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑎 ∈ V
8685, 5op2ndd 7686 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = ⟨𝑎, 𝑏⟩ → (2nd𝑦) = 𝑏)
8786breq2d 5045 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = ⟨𝑎, 𝑏⟩ → ((2nd𝑧) <N (2nd𝑦) ↔ (2nd𝑧) <N 𝑏))
8887notbid 321 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = ⟨𝑎, 𝑏⟩ → (¬ (2nd𝑧) <N (2nd𝑦) ↔ ¬ (2nd𝑧) <N 𝑏))
8984, 88imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑦)) ↔ (⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏)))
9089ralbidv 3165 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = ⟨𝑎, 𝑏⟩ → (∀𝑧 ∈ (N × N)(𝑦 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑦)) ↔ ∀𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏)))
91 df-nq 10327 . . . . . . . . . . . . . . . . . . . . . 22 Q = {𝑦 ∈ (N × N) ∣ ∀𝑧 ∈ (N × N)(𝑦 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑦))}
9290, 91elrab2 3634 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑎, 𝑏⟩ ∈ Q ↔ (⟨𝑎, 𝑏⟩ ∈ (N × N) ∧ ∀𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏)))
9392simplbi2 504 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑎, 𝑏⟩ ∈ (N × N) → (∀𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏) → ⟨𝑎, 𝑏⟩ ∈ Q))
9483, 93syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑎N𝑏N) → (∀𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏) → ⟨𝑎, 𝑏⟩ ∈ Q))
95 breq1 5036 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ~Q𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ ~Q𝑎, 𝑏⟩))
9695rspcev 3574 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑎, 𝑏⟩ ∈ Q ∧ ⟨𝑎, 𝑏⟩ ~Q𝑎, 𝑏⟩) → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)
9796expcom 417 . . . . . . . . . . . . . . . . . . 19 (⟨𝑎, 𝑏⟩ ~Q𝑎, 𝑏⟩ → (⟨𝑎, 𝑏⟩ ∈ Q → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
9882, 94, 97sylsyld 61 . . . . . . . . . . . . . . . . . 18 ((𝑎N𝑏N) → (∀𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏) → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
9998com12 32 . . . . . . . . . . . . . . . . 17 (∀𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏) → ((𝑎N𝑏N) → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
10099a1dd 50 . . . . . . . . . . . . . . . 16 (∀𝑧 ∈ (N × N)(⟨𝑎, 𝑏⟩ ~Q 𝑧 → ¬ (2nd𝑧) <N 𝑏) → ((𝑎N𝑏N) → (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
10178, 100pm2.61d2 184 . . . . . . . . . . . . . . 15 (∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) → ((𝑎N𝑏N) → (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
102101ralrimivv 3158 . . . . . . . . . . . . . 14 (∀𝑚𝑦𝑐N𝑑N (𝑑𝑚 → ∃𝑥Q 𝑥 ~Q𝑐, 𝑑⟩) → ∀𝑎N𝑏N (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
10324, 102sylbir 238 . . . . . . . . . . . . 13 (∀𝑚𝑦𝑎N𝑏N (𝑏𝑚 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩) → ∀𝑎N𝑏N (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
104103a1i 11 . . . . . . . . . . . 12 (𝑦 ∈ On → (∀𝑚𝑦𝑎N𝑏N (𝑏𝑚 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩) → ∀𝑎N𝑏N (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
10513, 104tfis2 7555 . . . . . . . . . . 11 (𝑦 ∈ On → ∀𝑎N𝑏N (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
106 rsp 3173 . . . . . . . . . . 11 (∀𝑎N𝑏N (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩) → (𝑎N → ∀𝑏N (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
107105, 106syl 17 . . . . . . . . . 10 (𝑦 ∈ On → (𝑎N → ∀𝑏N (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
108 rsp 3173 . . . . . . . . . 10 (∀𝑏N (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩) → (𝑏N → (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
109107, 108syl6 35 . . . . . . . . 9 (𝑦 ∈ On → (𝑎N → (𝑏N → (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))))
110109impd 414 . . . . . . . 8 (𝑦 ∈ On → ((𝑎N𝑏N) → (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
111110com12 32 . . . . . . 7 ((𝑎N𝑏N) → (𝑦 ∈ On → (𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)))
112111rexlimdv 3245 . . . . . 6 ((𝑎N𝑏N) → (∃𝑦 ∈ On 𝑏𝑦 → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
11310, 112mpd 15 . . . . 5 ((𝑎N𝑏N) → ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩)
114 breq2 5037 . . . . . 6 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑥 ~Q 𝐴𝑥 ~Q𝑎, 𝑏⟩))
115114rexbidv 3259 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → (∃𝑥Q 𝑥 ~Q 𝐴 ↔ ∃𝑥Q 𝑥 ~Q𝑎, 𝑏⟩))
116113, 115syl5ibrcom 250 . . . 4 ((𝑎N𝑏N) → (𝐴 = ⟨𝑎, 𝑏⟩ → ∃𝑥Q 𝑥 ~Q 𝐴))
117116rexlimivv 3254 . . 3 (∃𝑎N𝑏N 𝐴 = ⟨𝑎, 𝑏⟩ → ∃𝑥Q 𝑥 ~Q 𝐴)
1181, 117sylbi 220 . 2 (𝐴 ∈ (N × N) → ∃𝑥Q 𝑥 ~Q 𝐴)
119 breq2 5037 . . . . . 6 (𝑎 = 𝐴 → (𝑥 ~Q 𝑎𝑥 ~Q 𝐴))
120 breq2 5037 . . . . . 6 (𝑎 = 𝐴 → (𝑦 ~Q 𝑎𝑦 ~Q 𝐴))
121119, 120anbi12d 633 . . . . 5 (𝑎 = 𝐴 → ((𝑥 ~Q 𝑎𝑦 ~Q 𝑎) ↔ (𝑥 ~Q 𝐴𝑦 ~Q 𝐴)))
122121imbi1d 345 . . . 4 (𝑎 = 𝐴 → (((𝑥 ~Q 𝑎𝑦 ~Q 𝑎) → 𝑥 = 𝑦) ↔ ((𝑥 ~Q 𝐴𝑦 ~Q 𝐴) → 𝑥 = 𝑦)))
1231222ralbidv 3167 . . 3 (𝑎 = 𝐴 → (∀𝑥Q𝑦Q ((𝑥 ~Q 𝑎𝑦 ~Q 𝑎) → 𝑥 = 𝑦) ↔ ∀𝑥Q𝑦Q ((𝑥 ~Q 𝐴𝑦 ~Q 𝐴) → 𝑥 = 𝑦)))
12464a1i 11 . . . . . 6 ((𝑥 ~Q 𝑎𝑦 ~Q 𝑎) → ~Q Er (N × N))
125 simpl 486 . . . . . 6 ((𝑥 ~Q 𝑎𝑦 ~Q 𝑎) → 𝑥 ~Q 𝑎)
126 simpr 488 . . . . . 6 ((𝑥 ~Q 𝑎𝑦 ~Q 𝑎) → 𝑦 ~Q 𝑎)
127124, 125, 126ertr4d 8295 . . . . 5 ((𝑥 ~Q 𝑎𝑦 ~Q 𝑎) → 𝑥 ~Q 𝑦)
128 mulcompi 10311 . . . . . . . . . . 11 ((2nd𝑥) ·N (1st𝑥)) = ((1st𝑥) ·N (2nd𝑥))
129 elpqn 10340 . . . . . . . . . . . . . . 15 (𝑦Q𝑦 ∈ (N × N))
130 breq1 5036 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → (𝑦 ~Q 𝑧𝑥 ~Q 𝑧))
131 fveq2 6649 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑥 → (2nd𝑦) = (2nd𝑥))
132131breq2d 5045 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → ((2nd𝑧) <N (2nd𝑦) ↔ (2nd𝑧) <N (2nd𝑥)))
133132notbid 321 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → (¬ (2nd𝑧) <N (2nd𝑦) ↔ ¬ (2nd𝑧) <N (2nd𝑥)))
134130, 133imbi12d 348 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑥 → ((𝑦 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑦)) ↔ (𝑥 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑥))))
135134ralbidv 3165 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (∀𝑧 ∈ (N × N)(𝑦 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑦)) ↔ ∀𝑧 ∈ (N × N)(𝑥 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑥))))
136135, 91elrab2 3634 . . . . . . . . . . . . . . . 16 (𝑥Q ↔ (𝑥 ∈ (N × N) ∧ ∀𝑧 ∈ (N × N)(𝑥 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑥))))
137136simprbi 500 . . . . . . . . . . . . . . 15 (𝑥Q → ∀𝑧 ∈ (N × N)(𝑥 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑥)))
138 breq2 5037 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝑥 ~Q 𝑧𝑥 ~Q 𝑦))
139 fveq2 6649 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑦 → (2nd𝑧) = (2nd𝑦))
140139breq1d 5043 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → ((2nd𝑧) <N (2nd𝑥) ↔ (2nd𝑦) <N (2nd𝑥)))
141140notbid 321 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (¬ (2nd𝑧) <N (2nd𝑥) ↔ ¬ (2nd𝑦) <N (2nd𝑥)))
142138, 141imbi12d 348 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → ((𝑥 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑥)) ↔ (𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))))
143142rspcva 3572 . . . . . . . . . . . . . . 15 ((𝑦 ∈ (N × N) ∧ ∀𝑧 ∈ (N × N)(𝑥 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑥))) → (𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥)))
144129, 137, 143syl2anr 599 . . . . . . . . . . . . . 14 ((𝑥Q𝑦Q) → (𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥)))
145144imp 410 . . . . . . . . . . . . 13 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → ¬ (2nd𝑦) <N (2nd𝑥))
146 elpqn 10340 . . . . . . . . . . . . . . . . 17 (𝑥Q𝑥 ∈ (N × N))
14791rabeq2i 3438 . . . . . . . . . . . . . . . . . 18 (𝑦Q ↔ (𝑦 ∈ (N × N) ∧ ∀𝑧 ∈ (N × N)(𝑦 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑦))))
148147simprbi 500 . . . . . . . . . . . . . . . . 17 (𝑦Q → ∀𝑧 ∈ (N × N)(𝑦 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑦)))
149 breq2 5037 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑥 → (𝑦 ~Q 𝑧𝑦 ~Q 𝑥))
150 fveq2 6649 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑥 → (2nd𝑧) = (2nd𝑥))
151150breq1d 5043 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝑥 → ((2nd𝑧) <N (2nd𝑦) ↔ (2nd𝑥) <N (2nd𝑦)))
152151notbid 321 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑥 → (¬ (2nd𝑧) <N (2nd𝑦) ↔ ¬ (2nd𝑥) <N (2nd𝑦)))
153149, 152imbi12d 348 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → ((𝑦 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑦)) ↔ (𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦))))
154153rspcva 3572 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (N × N) ∧ ∀𝑧 ∈ (N × N)(𝑦 ~Q 𝑧 → ¬ (2nd𝑧) <N (2nd𝑦))) → (𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦)))
155146, 148, 154syl2an 598 . . . . . . . . . . . . . . . 16 ((𝑥Q𝑦Q) → (𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦)))
15664a1i 11 . . . . . . . . . . . . . . . . 17 (𝑥 ~Q 𝑦 → ~Q Er (N × N))
157 id 22 . . . . . . . . . . . . . . . . 17 (𝑥 ~Q 𝑦𝑥 ~Q 𝑦)
158156, 157ersym 8288 . . . . . . . . . . . . . . . 16 (𝑥 ~Q 𝑦𝑦 ~Q 𝑥)
159155, 158impel 509 . . . . . . . . . . . . . . 15 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → ¬ (2nd𝑥) <N (2nd𝑦))
160 xp2nd 7708 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
161146, 160syl 17 . . . . . . . . . . . . . . . . 17 (𝑥Q → (2nd𝑥) ∈ N)
162161ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → (2nd𝑥) ∈ N)
163 xp2nd 7708 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
164129, 163syl 17 . . . . . . . . . . . . . . . . 17 (𝑦Q → (2nd𝑦) ∈ N)
165164ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → (2nd𝑦) ∈ N)
166 ltsopi 10303 . . . . . . . . . . . . . . . . . . 19 <N Or N
167 sotric 5469 . . . . . . . . . . . . . . . . . . 19 (( <N Or N ∧ ((2nd𝑥) ∈ N ∧ (2nd𝑦) ∈ N)) → ((2nd𝑥) <N (2nd𝑦) ↔ ¬ ((2nd𝑥) = (2nd𝑦) ∨ (2nd𝑦) <N (2nd𝑥))))
168166, 167mpan 689 . . . . . . . . . . . . . . . . . 18 (((2nd𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((2nd𝑥) <N (2nd𝑦) ↔ ¬ ((2nd𝑥) = (2nd𝑦) ∨ (2nd𝑦) <N (2nd𝑥))))
169168notbid 321 . . . . . . . . . . . . . . . . 17 (((2nd𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → (¬ (2nd𝑥) <N (2nd𝑦) ↔ ¬ ¬ ((2nd𝑥) = (2nd𝑦) ∨ (2nd𝑦) <N (2nd𝑥))))
170 notnotb 318 . . . . . . . . . . . . . . . . 17 (((2nd𝑥) = (2nd𝑦) ∨ (2nd𝑦) <N (2nd𝑥)) ↔ ¬ ¬ ((2nd𝑥) = (2nd𝑦) ∨ (2nd𝑦) <N (2nd𝑥)))
171169, 170syl6bbr 292 . . . . . . . . . . . . . . . 16 (((2nd𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → (¬ (2nd𝑥) <N (2nd𝑦) ↔ ((2nd𝑥) = (2nd𝑦) ∨ (2nd𝑦) <N (2nd𝑥))))
172162, 165, 171syl2anc 587 . . . . . . . . . . . . . . 15 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → (¬ (2nd𝑥) <N (2nd𝑦) ↔ ((2nd𝑥) = (2nd𝑦) ∨ (2nd𝑦) <N (2nd𝑥))))
173159, 172mpbid 235 . . . . . . . . . . . . . 14 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → ((2nd𝑥) = (2nd𝑦) ∨ (2nd𝑦) <N (2nd𝑥)))
174173ord 861 . . . . . . . . . . . . 13 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → (¬ (2nd𝑥) = (2nd𝑦) → (2nd𝑦) <N (2nd𝑥)))
175145, 174mt3d 150 . . . . . . . . . . . 12 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → (2nd𝑥) = (2nd𝑦))
176175oveq2d 7155 . . . . . . . . . . 11 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → ((1st𝑥) ·N (2nd𝑥)) = ((1st𝑥) ·N (2nd𝑦)))
177128, 176syl5eq 2848 . . . . . . . . . 10 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → ((2nd𝑥) ·N (1st𝑥)) = ((1st𝑥) ·N (2nd𝑦)))
178 1st2nd2 7714 . . . . . . . . . . . . . 14 (𝑥 ∈ (N × N) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
179 1st2nd2 7714 . . . . . . . . . . . . . 14 (𝑦 ∈ (N × N) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
180178, 179breqan12d 5049 . . . . . . . . . . . . 13 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (𝑥 ~Q 𝑦 ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ~Q ⟨(1st𝑦), (2nd𝑦)⟩))
181 xp1st 7707 . . . . . . . . . . . . . . 15 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
182181, 160jca 515 . . . . . . . . . . . . . 14 (𝑥 ∈ (N × N) → ((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N))
183 xp1st 7707 . . . . . . . . . . . . . . 15 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
184183, 163jca 515 . . . . . . . . . . . . . 14 (𝑦 ∈ (N × N) → ((1st𝑦) ∈ N ∧ (2nd𝑦) ∈ N))
185 enqbreq 10334 . . . . . . . . . . . . . 14 ((((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N) ∧ ((1st𝑦) ∈ N ∧ (2nd𝑦) ∈ N)) → (⟨(1st𝑥), (2nd𝑥)⟩ ~Q ⟨(1st𝑦), (2nd𝑦)⟩ ↔ ((1st𝑥) ·N (2nd𝑦)) = ((2nd𝑥) ·N (1st𝑦))))
186182, 184, 185syl2an 598 . . . . . . . . . . . . 13 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (⟨(1st𝑥), (2nd𝑥)⟩ ~Q ⟨(1st𝑦), (2nd𝑦)⟩ ↔ ((1st𝑥) ·N (2nd𝑦)) = ((2nd𝑥) ·N (1st𝑦))))
187180, 186bitrd 282 . . . . . . . . . . . 12 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (𝑥 ~Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((2nd𝑥) ·N (1st𝑦))))
188146, 129, 187syl2an 598 . . . . . . . . . . 11 ((𝑥Q𝑦Q) → (𝑥 ~Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((2nd𝑥) ·N (1st𝑦))))
189188biimpa 480 . . . . . . . . . 10 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → ((1st𝑥) ·N (2nd𝑦)) = ((2nd𝑥) ·N (1st𝑦)))
190177, 189eqtrd 2836 . . . . . . . . 9 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → ((2nd𝑥) ·N (1st𝑥)) = ((2nd𝑥) ·N (1st𝑦)))
191146ad2antrr 725 . . . . . . . . . 10 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → 𝑥 ∈ (N × N))
192 mulcanpi 10315 . . . . . . . . . . 11 (((2nd𝑥) ∈ N ∧ (1st𝑥) ∈ N) → (((2nd𝑥) ·N (1st𝑥)) = ((2nd𝑥) ·N (1st𝑦)) ↔ (1st𝑥) = (1st𝑦)))
193160, 181, 192syl2anc 587 . . . . . . . . . 10 (𝑥 ∈ (N × N) → (((2nd𝑥) ·N (1st𝑥)) = ((2nd𝑥) ·N (1st𝑦)) ↔ (1st𝑥) = (1st𝑦)))
194191, 193syl 17 . . . . . . . . 9 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → (((2nd𝑥) ·N (1st𝑥)) = ((2nd𝑥) ·N (1st𝑦)) ↔ (1st𝑥) = (1st𝑦)))
195190, 194mpbid 235 . . . . . . . 8 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → (1st𝑥) = (1st𝑦))
196195, 175opeq12d 4776 . . . . . . 7 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
197191, 178syl 17 . . . . . . 7 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
198129ad2antlr 726 . . . . . . . 8 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → 𝑦 ∈ (N × N))
199198, 179syl 17 . . . . . . 7 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
200196, 197, 1993eqtr4d 2846 . . . . . 6 (((𝑥Q𝑦Q) ∧ 𝑥 ~Q 𝑦) → 𝑥 = 𝑦)
201200ex 416 . . . . 5 ((𝑥Q𝑦Q) → (𝑥 ~Q 𝑦𝑥 = 𝑦))
202127, 201syl5 34 . . . 4 ((𝑥Q𝑦Q) → ((𝑥 ~Q 𝑎𝑦 ~Q 𝑎) → 𝑥 = 𝑦))
203202rgen2 3171 . . 3 𝑥Q𝑦Q ((𝑥 ~Q 𝑎𝑦 ~Q 𝑎) → 𝑥 = 𝑦)
204123, 203vtoclg 3518 . 2 (𝐴 ∈ (N × N) → ∀𝑥Q𝑦Q ((𝑥 ~Q 𝐴𝑦 ~Q 𝐴) → 𝑥 = 𝑦))
205 breq1 5036 . . 3 (𝑥 = 𝑦 → (𝑥 ~Q 𝐴𝑦 ~Q 𝐴))
206205reu4 3673 . 2 (∃!𝑥Q 𝑥 ~Q 𝐴 ↔ (∃𝑥Q 𝑥 ~Q 𝐴 ∧ ∀𝑥Q𝑦Q ((𝑥 ~Q 𝐴𝑦 ~Q 𝐴) → 𝑥 = 𝑦)))
207118, 204, 206sylanbrc 586 1 (𝐴 ∈ (N × N) → ∃!𝑥Q 𝑥 ~Q 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wrex 3110  ∃!wreu 3111  cop 4534   class class class wbr 5033   Or wor 5441   × cxp 5521  Oncon0 6163  suc csuc 6165  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674   Er wer 8273  Ncnpi 10259   ·N cmi 10261   <N clti 10262   ~Q ceq 10266  Qcnq 10267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-oadd 8093  df-omul 8094  df-er 8276  df-ni 10287  df-mi 10289  df-lti 10290  df-enq 10326  df-nq 10327
This theorem is referenced by:  nqerf  10345  enqeq  10349
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