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Theorem unxpdomlem3 8708
Description: Lemma for unxpdom 8709. (Contributed by Mario Carneiro, 13-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
Hypotheses
Ref Expression
unxpdomlem1.1 𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)
unxpdomlem1.2 𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
Assertion
Ref Expression
unxpdomlem3 ((1o𝑎 ∧ 1o𝑏) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
Distinct variable group:   𝑎,𝑏,𝑚,𝑛,𝑠,𝑡,𝑥
Allowed substitution hints:   𝐹(𝑥,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)   𝐺(𝑥,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)

Proof of Theorem unxpdomlem3
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1sdom 8705 . . 3 (𝑎 ∈ V → (1o𝑎 ↔ ∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛))
21elv 3446 . 2 (1o𝑎 ↔ ∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛)
3 1sdom 8705 . . 3 (𝑏 ∈ V → (1o𝑏 ↔ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡))
43elv 3446 . 2 (1o𝑏 ↔ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡)
5 reeanv 3320 . . 3 (∃𝑚𝑎𝑠𝑏 (∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡) ↔ (∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡))
6 reeanv 3320 . . . . 5 (∃𝑛𝑎𝑡𝑏𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ↔ (∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡))
7 vex 3444 . . . . . . . . 9 𝑎 ∈ V
8 vex 3444 . . . . . . . . 9 𝑏 ∈ V
97, 8unex 7449 . . . . . . . 8 (𝑎𝑏) ∈ V
107, 8xpex 7456 . . . . . . . 8 (𝑎 × 𝑏) ∈ V
11 unxpdomlem1.2 . . . . . . . . . . 11 𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
12 simpr 488 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ 𝑥𝑎) → 𝑥𝑎)
13 simp2r 1197 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑡𝑏)
14 simp1r 1195 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑠𝑏)
1513, 14ifcld 4470 . . . . . . . . . . . . . 14 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → if(𝑥 = 𝑚, 𝑡, 𝑠) ∈ 𝑏)
1615ad2antrr 725 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ 𝑥𝑎) → if(𝑥 = 𝑚, 𝑡, 𝑠) ∈ 𝑏)
1712, 16opelxpd 5557 . . . . . . . . . . . 12 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ 𝑥𝑎) → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ ∈ (𝑎 × 𝑏))
18 simp2l 1196 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑛𝑎)
19 simp1l 1194 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑚𝑎)
2018, 19ifcld 4470 . . . . . . . . . . . . . 14 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → if(𝑥 = 𝑡, 𝑛, 𝑚) ∈ 𝑎)
2120ad2antrr 725 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ ¬ 𝑥𝑎) → if(𝑥 = 𝑡, 𝑛, 𝑚) ∈ 𝑎)
22 simpr 488 . . . . . . . . . . . . . . 15 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → 𝑥 ∈ (𝑎𝑏))
23 elun 4076 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝑎𝑏) ↔ (𝑥𝑎𝑥𝑏))
2422, 23sylib 221 . . . . . . . . . . . . . 14 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → (𝑥𝑎𝑥𝑏))
2524orcanai 1000 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ ¬ 𝑥𝑎) → 𝑥𝑏)
2621, 25opelxpd 5557 . . . . . . . . . . . 12 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ ¬ 𝑥𝑎) → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ ∈ (𝑎 × 𝑏))
2717, 26ifclda 4459 . . . . . . . . . . 11 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩) ∈ (𝑎 × 𝑏))
2811, 27eqeltrid 2894 . . . . . . . . . 10 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → 𝐺 ∈ (𝑎 × 𝑏))
29 unxpdomlem1.1 . . . . . . . . . 10 𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)
3028, 29fmptd 6855 . . . . . . . . 9 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝐹:(𝑎𝑏)⟶(𝑎 × 𝑏))
3129, 11unxpdomlem1 8706 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑎𝑏) → (𝐹𝑧) = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
3231ad2antrl 727 . . . . . . . . . . . . . . 15 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → (𝐹𝑧) = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
33 iftrue 4431 . . . . . . . . . . . . . . . 16 (𝑧𝑎 → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
3433adantr 484 . . . . . . . . . . . . . . 15 ((𝑧𝑎𝑤𝑎) → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
3532, 34sylan9eq 2853 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → (𝐹𝑧) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
3629, 11unxpdomlem1 8706 . . . . . . . . . . . . . . . 16 (𝑤 ∈ (𝑎𝑏) → (𝐹𝑤) = if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
3736ad2antll 728 . . . . . . . . . . . . . . 15 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → (𝐹𝑤) = if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
38 iftrue 4431 . . . . . . . . . . . . . . . 16 (𝑤𝑎 → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
3938adantl 485 . . . . . . . . . . . . . . 15 ((𝑧𝑎𝑤𝑎) → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
4037, 39sylan9eq 2853 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → (𝐹𝑤) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
4135, 40eqeq12d 2814 . . . . . . . . . . . . 13 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩))
42 vex 3444 . . . . . . . . . . . . . 14 𝑧 ∈ V
43 vex 3444 . . . . . . . . . . . . . . 15 𝑡 ∈ V
44 vex 3444 . . . . . . . . . . . . . . 15 𝑠 ∈ V
4543, 44ifex 4473 . . . . . . . . . . . . . 14 if(𝑧 = 𝑚, 𝑡, 𝑠) ∈ V
4642, 45opth1 5332 . . . . . . . . . . . . 13 (⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩ → 𝑧 = 𝑤)
4741, 46syl6bi 256 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
48 simprr 772 . . . . . . . . . . . . . 14 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → 𝑤 ∈ (𝑎𝑏))
49 simpll 766 . . . . . . . . . . . . . 14 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → ¬ 𝑚 = 𝑛)
50 simplr 768 . . . . . . . . . . . . . 14 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → ¬ 𝑠 = 𝑡)
5129, 11, 48, 49, 50unxpdomlem2 8707 . . . . . . . . . . . . 13 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ¬ (𝐹𝑧) = (𝐹𝑤))
5251pm2.21d 121 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
53 eqcom 2805 . . . . . . . . . . . . 13 ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑤) = (𝐹𝑧))
54 simprl 770 . . . . . . . . . . . . . . . 16 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → 𝑧 ∈ (𝑎𝑏))
5529, 11, 54, 49, 50unxpdomlem2 8707 . . . . . . . . . . . . . . 15 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑤𝑎 ∧ ¬ 𝑧𝑎)) → ¬ (𝐹𝑤) = (𝐹𝑧))
5655ancom2s 649 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎𝑤𝑎)) → ¬ (𝐹𝑤) = (𝐹𝑧))
5756pm2.21d 121 . . . . . . . . . . . . 13 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎𝑤𝑎)) → ((𝐹𝑤) = (𝐹𝑧) → 𝑧 = 𝑤))
5853, 57syl5bi 245 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
59 iffalse 4434 . . . . . . . . . . . . . . . 16 𝑧𝑎 → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
6059adantr 484 . . . . . . . . . . . . . . 15 ((¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎) → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
6132, 60sylan9eq 2853 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑧) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
62 iffalse 4434 . . . . . . . . . . . . . . . 16 𝑤𝑎 → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
6362adantl 485 . . . . . . . . . . . . . . 15 ((¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎) → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
6437, 63sylan9eq 2853 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑤) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
6561, 64eqeq12d 2814 . . . . . . . . . . . . 13 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
66 vex 3444 . . . . . . . . . . . . . . . 16 𝑛 ∈ V
67 vex 3444 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
6866, 67ifex 4473 . . . . . . . . . . . . . . 15 if(𝑧 = 𝑡, 𝑛, 𝑚) ∈ V
6968, 42opth 5333 . . . . . . . . . . . . . 14 (⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩ ↔ (if(𝑧 = 𝑡, 𝑛, 𝑚) = if(𝑤 = 𝑡, 𝑛, 𝑚) ∧ 𝑧 = 𝑤))
7069simprbi 500 . . . . . . . . . . . . 13 (⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩ → 𝑧 = 𝑤)
7165, 70syl6bi 256 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
7247, 52, 58, 714casesdan 1037 . . . . . . . . . . 11 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
7372ralrimivva 3156 . . . . . . . . . 10 ((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → ∀𝑧 ∈ (𝑎𝑏)∀𝑤 ∈ (𝑎𝑏)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
74733ad2ant3 1132 . . . . . . . . 9 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → ∀𝑧 ∈ (𝑎𝑏)∀𝑤 ∈ (𝑎𝑏)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
75 dff13 6991 . . . . . . . . 9 (𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏) ↔ (𝐹:(𝑎𝑏)⟶(𝑎 × 𝑏) ∧ ∀𝑧 ∈ (𝑎𝑏)∀𝑤 ∈ (𝑎𝑏)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
7630, 74, 75sylanbrc 586 . . . . . . . 8 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏))
77 f1dom2g 8510 . . . . . . . 8 (((𝑎𝑏) ∈ V ∧ (𝑎 × 𝑏) ∈ V ∧ 𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏)) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
789, 10, 76, 77mp3an12i 1462 . . . . . . 7 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
79783expia 1118 . . . . . 6 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏)) → ((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏)))
8079rexlimdvva 3253 . . . . 5 ((𝑚𝑎𝑠𝑏) → (∃𝑛𝑎𝑡𝑏𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏)))
816, 80syl5bir 246 . . . 4 ((𝑚𝑎𝑠𝑏) → ((∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏)))
8281rexlimivv 3251 . . 3 (∃𝑚𝑎𝑠𝑏 (∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
835, 82sylbir 238 . 2 ((∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
842, 4, 83syl2anb 600 1 ((1o𝑎 ∧ 1o𝑏) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2111  wral 3106  wrex 3107  Vcvv 3441  cun 3879  ifcif 4425  cop 4531   class class class wbr 5030  cmpt 5110   × cxp 5517  wf 6320  1-1wf1 6321  cfv 6324  1oc1o 8078  cdom 8490  csdm 8491
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1o 8085  df-2o 8086  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495
This theorem is referenced by:  unxpdom  8709
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