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Theorem unxpdomlem3 9252
Description: Lemma for unxpdom 9253. (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 9248 . . 3 (𝑎 ∈ V → (1o𝑎 ↔ ∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛))
21elv 3481 . 2 (1o𝑎 ↔ ∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛)
3 1sdom 9248 . . 3 (𝑏 ∈ V → (1o𝑏 ↔ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡))
43elv 3481 . 2 (1o𝑏 ↔ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡)
5 reeanv 3227 . . 3 (∃𝑚𝑎𝑠𝑏 (∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡) ↔ (∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡))
6 reeanv 3227 . . . . 5 (∃𝑛𝑎𝑡𝑏𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ↔ (∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡))
7 vex 3479 . . . . . . . . 9 𝑎 ∈ V
8 vex 3479 . . . . . . . . 9 𝑏 ∈ V
97, 8unex 7733 . . . . . . . 8 (𝑎𝑏) ∈ V
107, 8xpex 7740 . . . . . . . 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 486 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ 𝑥𝑎) → 𝑥𝑎)
13 simp2r 1201 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑡𝑏)
14 simp1r 1199 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑠𝑏)
1513, 14ifcld 4575 . . . . . . . . . . . . . 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 5716 . . . . . . . . . . . 12 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ 𝑥𝑎) → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ ∈ (𝑎 × 𝑏))
18 simp2l 1200 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑛𝑎)
19 simp1l 1198 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑚𝑎)
2018, 19ifcld 4575 . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . 15 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → 𝑥 ∈ (𝑎𝑏))
23 elun 4149 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝑎𝑏) ↔ (𝑥𝑎𝑥𝑏))
2422, 23sylib 217 . . . . . . . . . . . . . 14 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → (𝑥𝑎𝑥𝑏))
2524orcanai 1002 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ ¬ 𝑥𝑎) → 𝑥𝑏)
2621, 25opelxpd 5716 . . . . . . . . . . . 12 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ ¬ 𝑥𝑎) → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ ∈ (𝑎 × 𝑏))
2717, 26ifclda 4564 . . . . . . . . . . 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 2838 . . . . . . . . . 10 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → 𝐺 ∈ (𝑎 × 𝑏))
29 unxpdomlem1.1 . . . . . . . . . 10 𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)
3028, 29fmptd 7114 . . . . . . . . 9 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝐹:(𝑎𝑏)⟶(𝑎 × 𝑏))
3129, 11unxpdomlem1 9250 . . . . . . . . . . . . . . . 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 4535 . . . . . . . . . . . . . . . 16 (𝑧𝑎 → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
3433adantr 482 . . . . . . . . . . . . . . 15 ((𝑧𝑎𝑤𝑎) → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
3532, 34sylan9eq 2793 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → (𝐹𝑧) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
3629, 11unxpdomlem1 9250 . . . . . . . . . . . . . . . 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 4535 . . . . . . . . . . . . . . . 16 (𝑤𝑎 → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
3938adantl 483 . . . . . . . . . . . . . . 15 ((𝑧𝑎𝑤𝑎) → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
4037, 39sylan9eq 2793 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → (𝐹𝑤) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
4135, 40eqeq12d 2749 . . . . . . . . . . . . 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 3479 . . . . . . . . . . . . . 14 𝑧 ∈ V
43 vex 3479 . . . . . . . . . . . . . . 15 𝑡 ∈ V
44 vex 3479 . . . . . . . . . . . . . . 15 𝑠 ∈ V
4543, 44ifex 4579 . . . . . . . . . . . . . 14 if(𝑧 = 𝑚, 𝑡, 𝑠) ∈ V
4642, 45opth1 5476 . . . . . . . . . . . . 13 (⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩ → 𝑧 = 𝑤)
4741, 46syl6bi 253 . . . . . . . . . . . 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 9251 . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . 13 ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑤) = (𝐹𝑧))
54 simprl 770 . . . . . . . . . . . . . . . 16 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → 𝑧 ∈ (𝑎𝑏))
5529, 11, 54, 49, 50unxpdomlem2 9251 . . . . . . . . . . . . . . 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, 57biimtrid 241 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
59 iffalse 4538 . . . . . . . . . . . . . . . 16 𝑧𝑎 → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
6059adantr 482 . . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑧) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
62 iffalse 4538 . . . . . . . . . . . . . . . 16 𝑤𝑎 → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
6362adantl 483 . . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑤) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
6561, 64eqeq12d 2749 . . . . . . . . . . . . 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 3479 . . . . . . . . . . . . . . . 16 𝑛 ∈ V
67 vex 3479 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
6866, 67ifex 4579 . . . . . . . . . . . . . . 15 if(𝑧 = 𝑡, 𝑛, 𝑚) ∈ V
6968, 42opth 5477 . . . . . . . . . . . . . 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 498 . . . . . . . . . . . . 13 (⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩ → 𝑧 = 𝑤)
7165, 70syl6bi 253 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
7247, 52, 58, 714casesdan 1041 . . . . . . . . . . 11 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
7372ralrimivva 3201 . . . . . . . . . 10 ((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → ∀𝑧 ∈ (𝑎𝑏)∀𝑤 ∈ (𝑎𝑏)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
74733ad2ant3 1136 . . . . . . . . 9 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → ∀𝑧 ∈ (𝑎𝑏)∀𝑤 ∈ (𝑎𝑏)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
75 dff13 7254 . . . . . . . . 9 (𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏) ↔ (𝐹:(𝑎𝑏)⟶(𝑎 × 𝑏) ∧ ∀𝑧 ∈ (𝑎𝑏)∀𝑤 ∈ (𝑎𝑏)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
7630, 74, 75sylanbrc 584 . . . . . . . 8 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏))
77 f1dom2g 8965 . . . . . . . 8 (((𝑎𝑏) ∈ V ∧ (𝑎 × 𝑏) ∈ V ∧ 𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏)) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
789, 10, 76, 77mp3an12i 1466 . . . . . . 7 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
79783expia 1122 . . . . . 6 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏)) → ((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏)))
8079rexlimdvva 3212 . . . . 5 ((𝑚𝑎𝑠𝑏) → (∃𝑛𝑎𝑡𝑏𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏)))
816, 80biimtrrid 242 . . . 4 ((𝑚𝑎𝑠𝑏) → ((∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏)))
8281rexlimivv 3200 . . 3 (∃𝑚𝑎𝑠𝑏 (∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
835, 82sylbir 234 . 2 ((∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
842, 4, 83syl2anb 599 1 ((1o𝑎 ∧ 1o𝑏) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  Vcvv 3475  cun 3947  ifcif 4529  cop 4635   class class class wbr 5149  cmpt 5232   × cxp 5675  wf 6540  1-1wf1 6541  cfv 6544  1oc1o 8459  cdom 8937  csdm 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1o 8466  df-2o 8467  df-en 8940  df-dom 8941  df-sdom 8942
This theorem is referenced by:  unxpdom  9253
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