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Theorem nqpr 9780
 Description: The canonical embedding of the rationals into the reals. (Contributed by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
nqpr (𝐴Q → {𝑥𝑥 <Q 𝐴} ∈ P)
Distinct variable group:   𝑥,𝐴

Proof of Theorem nqpr
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nsmallnq 9743 . . . . 5 (𝐴Q → ∃𝑥 𝑥 <Q 𝐴)
2 abn0 3928 . . . . 5 ({𝑥𝑥 <Q 𝐴} ≠ ∅ ↔ ∃𝑥 𝑥 <Q 𝐴)
31, 2sylibr 224 . . . 4 (𝐴Q → {𝑥𝑥 <Q 𝐴} ≠ ∅)
4 0pss 3985 . . . 4 (∅ ⊊ {𝑥𝑥 <Q 𝐴} ↔ {𝑥𝑥 <Q 𝐴} ≠ ∅)
53, 4sylibr 224 . . 3 (𝐴Q → ∅ ⊊ {𝑥𝑥 <Q 𝐴})
6 ltrelnq 9692 . . . . . . 7 <Q ⊆ (Q × Q)
76brel 5128 . . . . . 6 (𝑥 <Q 𝐴 → (𝑥Q𝐴Q))
87simpld 475 . . . . 5 (𝑥 <Q 𝐴𝑥Q)
98abssi 3656 . . . 4 {𝑥𝑥 <Q 𝐴} ⊆ Q
10 ltsonq 9735 . . . . . . 7 <Q Or Q
1110, 6soirri 5481 . . . . . 6 ¬ 𝐴 <Q 𝐴
12 breq1 4616 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 <Q 𝐴𝐴 <Q 𝐴))
1312elabg 3334 . . . . . 6 (𝐴Q → (𝐴 ∈ {𝑥𝑥 <Q 𝐴} ↔ 𝐴 <Q 𝐴))
1411, 13mtbiri 317 . . . . 5 (𝐴Q → ¬ 𝐴 ∈ {𝑥𝑥 <Q 𝐴})
1514ancli 573 . . . 4 (𝐴Q → (𝐴Q ∧ ¬ 𝐴 ∈ {𝑥𝑥 <Q 𝐴}))
16 ssnelpss 3696 . . . 4 ({𝑥𝑥 <Q 𝐴} ⊆ Q → ((𝐴Q ∧ ¬ 𝐴 ∈ {𝑥𝑥 <Q 𝐴}) → {𝑥𝑥 <Q 𝐴} ⊊ Q))
179, 15, 16mpsyl 68 . . 3 (𝐴Q → {𝑥𝑥 <Q 𝐴} ⊊ Q)
185, 17jca 554 . 2 (𝐴Q → (∅ ⊊ {𝑥𝑥 <Q 𝐴} ∧ {𝑥𝑥 <Q 𝐴} ⊊ Q))
19 vex 3189 . . . . 5 𝑦 ∈ V
20 breq1 4616 . . . . 5 (𝑥 = 𝑦 → (𝑥 <Q 𝐴𝑦 <Q 𝐴))
2119, 20elab 3333 . . . 4 (𝑦 ∈ {𝑥𝑥 <Q 𝐴} ↔ 𝑦 <Q 𝐴)
2210, 6sotri 5482 . . . . . . . . 9 ((𝑧 <Q 𝑦𝑦 <Q 𝐴) → 𝑧 <Q 𝐴)
2322expcom 451 . . . . . . . 8 (𝑦 <Q 𝐴 → (𝑧 <Q 𝑦𝑧 <Q 𝐴))
2423adantl 482 . . . . . . 7 ((𝐴Q𝑦 <Q 𝐴) → (𝑧 <Q 𝑦𝑧 <Q 𝐴))
25 vex 3189 . . . . . . . 8 𝑧 ∈ V
26 breq1 4616 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥 <Q 𝐴𝑧 <Q 𝐴))
2725, 26elab 3333 . . . . . . 7 (𝑧 ∈ {𝑥𝑥 <Q 𝐴} ↔ 𝑧 <Q 𝐴)
2824, 27syl6ibr 242 . . . . . 6 ((𝐴Q𝑦 <Q 𝐴) → (𝑧 <Q 𝑦𝑧 ∈ {𝑥𝑥 <Q 𝐴}))
2928alrimiv 1852 . . . . 5 ((𝐴Q𝑦 <Q 𝐴) → ∀𝑧(𝑧 <Q 𝑦𝑧 ∈ {𝑥𝑥 <Q 𝐴}))
30 ltbtwnnq 9744 . . . . . . . 8 (𝑦 <Q 𝐴 ↔ ∃𝑧(𝑦 <Q 𝑧𝑧 <Q 𝐴))
3127anbi2i 729 . . . . . . . . . . 11 ((𝑦 <Q 𝑧𝑧 ∈ {𝑥𝑥 <Q 𝐴}) ↔ (𝑦 <Q 𝑧𝑧 <Q 𝐴))
3231biimpri 218 . . . . . . . . . 10 ((𝑦 <Q 𝑧𝑧 <Q 𝐴) → (𝑦 <Q 𝑧𝑧 ∈ {𝑥𝑥 <Q 𝐴}))
3332ancomd 467 . . . . . . . . 9 ((𝑦 <Q 𝑧𝑧 <Q 𝐴) → (𝑧 ∈ {𝑥𝑥 <Q 𝐴} ∧ 𝑦 <Q 𝑧))
3433eximi 1759 . . . . . . . 8 (∃𝑧(𝑦 <Q 𝑧𝑧 <Q 𝐴) → ∃𝑧(𝑧 ∈ {𝑥𝑥 <Q 𝐴} ∧ 𝑦 <Q 𝑧))
3530, 34sylbi 207 . . . . . . 7 (𝑦 <Q 𝐴 → ∃𝑧(𝑧 ∈ {𝑥𝑥 <Q 𝐴} ∧ 𝑦 <Q 𝑧))
3635adantl 482 . . . . . 6 ((𝐴Q𝑦 <Q 𝐴) → ∃𝑧(𝑧 ∈ {𝑥𝑥 <Q 𝐴} ∧ 𝑦 <Q 𝑧))
37 df-rex 2913 . . . . . 6 (∃𝑧 ∈ {𝑥𝑥 <Q 𝐴}𝑦 <Q 𝑧 ↔ ∃𝑧(𝑧 ∈ {𝑥𝑥 <Q 𝐴} ∧ 𝑦 <Q 𝑧))
3836, 37sylibr 224 . . . . 5 ((𝐴Q𝑦 <Q 𝐴) → ∃𝑧 ∈ {𝑥𝑥 <Q 𝐴}𝑦 <Q 𝑧)
3929, 38jca 554 . . . 4 ((𝐴Q𝑦 <Q 𝐴) → (∀𝑧(𝑧 <Q 𝑦𝑧 ∈ {𝑥𝑥 <Q 𝐴}) ∧ ∃𝑧 ∈ {𝑥𝑥 <Q 𝐴}𝑦 <Q 𝑧))
4021, 39sylan2b 492 . . 3 ((𝐴Q𝑦 ∈ {𝑥𝑥 <Q 𝐴}) → (∀𝑧(𝑧 <Q 𝑦𝑧 ∈ {𝑥𝑥 <Q 𝐴}) ∧ ∃𝑧 ∈ {𝑥𝑥 <Q 𝐴}𝑦 <Q 𝑧))
4140ralrimiva 2960 . 2 (𝐴Q → ∀𝑦 ∈ {𝑥𝑥 <Q 𝐴} (∀𝑧(𝑧 <Q 𝑦𝑧 ∈ {𝑥𝑥 <Q 𝐴}) ∧ ∃𝑧 ∈ {𝑥𝑥 <Q 𝐴}𝑦 <Q 𝑧))
42 elnp 9753 . 2 ({𝑥𝑥 <Q 𝐴} ∈ P ↔ ((∅ ⊊ {𝑥𝑥 <Q 𝐴} ∧ {𝑥𝑥 <Q 𝐴} ⊊ Q) ∧ ∀𝑦 ∈ {𝑥𝑥 <Q 𝐴} (∀𝑧(𝑧 <Q 𝑦𝑧 ∈ {𝑥𝑥 <Q 𝐴}) ∧ ∃𝑧 ∈ {𝑥𝑥 <Q 𝐴}𝑦 <Q 𝑧)))
4318, 41, 42sylanbrc 697 1 (𝐴Q → {𝑥𝑥 <Q 𝐴} ∈ P)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1478  ∃wex 1701   ∈ wcel 1987  {cab 2607   ≠ wne 2790  ∀wral 2907  ∃wrex 2908   ⊆ wss 3555   ⊊ wpss 3556  ∅c0 3891   class class class wbr 4613  Qcnq 9618
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