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Theorem unxpdomlem3 8110
Description: Lemma for unxpdom 8111. (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 vex 3189 . . 3 𝑎 ∈ V
2 1sdom 8107 . . 3 (𝑎 ∈ V → (1𝑜𝑎 ↔ ∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛))
31, 2ax-mp 5 . 2 (1𝑜𝑎 ↔ ∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛)
4 vex 3189 . . 3 𝑏 ∈ V
5 1sdom 8107 . . 3 (𝑏 ∈ V → (1𝑜𝑏 ↔ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡))
64, 5ax-mp 5 . 2 (1𝑜𝑏 ↔ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡)
7 reeanv 3097 . . 3 (∃𝑚𝑎𝑠𝑏 (∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡) ↔ (∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡))
8 reeanv 3097 . . . . 5 (∃𝑛𝑎𝑡𝑏𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ↔ (∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡))
9 unxpdomlem1.2 . . . . . . . . . . 11 𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
10 simpr 477 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ 𝑥𝑎) → 𝑥𝑎)
11 simp2r 1086 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑡𝑏)
12 simp1r 1084 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑠𝑏)
1311, 12ifcld 4103 . . . . . . . . . . . . . 14 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → if(𝑥 = 𝑚, 𝑡, 𝑠) ∈ 𝑏)
1413ad2antrr 761 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ 𝑥𝑎) → if(𝑥 = 𝑚, 𝑡, 𝑠) ∈ 𝑏)
15 opelxpi 5108 . . . . . . . . . . . . 13 ((𝑥𝑎 ∧ if(𝑥 = 𝑚, 𝑡, 𝑠) ∈ 𝑏) → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ ∈ (𝑎 × 𝑏))
1610, 14, 15syl2anc 692 . . . . . . . . . . . 12 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ 𝑥𝑎) → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ ∈ (𝑎 × 𝑏))
17 simp2l 1085 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑛𝑎)
18 simp1l 1083 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑚𝑎)
1917, 18ifcld 4103 . . . . . . . . . . . . . 14 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → if(𝑥 = 𝑡, 𝑛, 𝑚) ∈ 𝑎)
2019ad2antrr 761 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ ¬ 𝑥𝑎) → if(𝑥 = 𝑡, 𝑛, 𝑚) ∈ 𝑎)
21 simpr 477 . . . . . . . . . . . . . . 15 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → 𝑥 ∈ (𝑎𝑏))
22 elun 3731 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝑎𝑏) ↔ (𝑥𝑎𝑥𝑏))
2321, 22sylib 208 . . . . . . . . . . . . . 14 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → (𝑥𝑎𝑥𝑏))
2423orcanai 951 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ ¬ 𝑥𝑎) → 𝑥𝑏)
25 opelxpi 5108 . . . . . . . . . . . . 13 ((if(𝑥 = 𝑡, 𝑛, 𝑚) ∈ 𝑎𝑥𝑏) → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ ∈ (𝑎 × 𝑏))
2620, 24, 25syl2anc 692 . . . . . . . . . . . 12 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ ¬ 𝑥𝑎) → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ ∈ (𝑎 × 𝑏))
2716, 26ifclda 4092 . . . . . . . . . . 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))
289, 27syl5eqel 2702 . . . . . . . . . 10 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → 𝐺 ∈ (𝑎 × 𝑏))
29 unxpdomlem1.1 . . . . . . . . . 10 𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)
3028, 29fmptd 6340 . . . . . . . . 9 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝐹:(𝑎𝑏)⟶(𝑎 × 𝑏))
3129, 9unxpdomlem1 8108 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑎𝑏) → (𝐹𝑧) = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
3231ad2antrl 763 . . . . . . . . . . . . . . 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 4064 . . . . . . . . . . . . . . . 16 (𝑧𝑎 → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
3433adantr 481 . . . . . . . . . . . . . . 15 ((𝑧𝑎𝑤𝑎) → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
3532, 34sylan9eq 2675 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → (𝐹𝑧) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
3629, 9unxpdomlem1 8108 . . . . . . . . . . . . . . . 16 (𝑤 ∈ (𝑎𝑏) → (𝐹𝑤) = if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
3736ad2antll 764 . . . . . . . . . . . . . . 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 4064 . . . . . . . . . . . . . . . 16 (𝑤𝑎 → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
3938adantl 482 . . . . . . . . . . . . . . 15 ((𝑧𝑎𝑤𝑎) → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
4037, 39sylan9eq 2675 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → (𝐹𝑤) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
4135, 40eqeq12d 2636 . . . . . . . . . . . . 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 3189 . . . . . . . . . . . . . 14 𝑧 ∈ V
43 vex 3189 . . . . . . . . . . . . . . 15 𝑡 ∈ V
44 vex 3189 . . . . . . . . . . . . . . 15 𝑠 ∈ V
4543, 44ifex 4128 . . . . . . . . . . . . . 14 if(𝑧 = 𝑚, 𝑡, 𝑠) ∈ V
4642, 45opth1 4904 . . . . . . . . . . . . 13 (⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩ → 𝑧 = 𝑤)
4741, 46syl6bi 243 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
48 simprr 795 . . . . . . . . . . . . . 14 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → 𝑤 ∈ (𝑎𝑏))
49 simpll 789 . . . . . . . . . . . . . 14 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → ¬ 𝑚 = 𝑛)
50 simplr 791 . . . . . . . . . . . . . 14 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → ¬ 𝑠 = 𝑡)
5129, 9, 48, 49, 50unxpdomlem2 8109 . . . . . . . . . . . . 13 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ¬ (𝐹𝑧) = (𝐹𝑤))
5251pm2.21d 118 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
53 eqcom 2628 . . . . . . . . . . . . 13 ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑤) = (𝐹𝑧))
54 simprl 793 . . . . . . . . . . . . . . . 16 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → 𝑧 ∈ (𝑎𝑏))
5529, 9, 54, 49, 50unxpdomlem2 8109 . . . . . . . . . . . . . . 15 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑤𝑎 ∧ ¬ 𝑧𝑎)) → ¬ (𝐹𝑤) = (𝐹𝑧))
5655ancom2s 843 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎𝑤𝑎)) → ¬ (𝐹𝑤) = (𝐹𝑧))
5756pm2.21d 118 . . . . . . . . . . . . 13 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎𝑤𝑎)) → ((𝐹𝑤) = (𝐹𝑧) → 𝑧 = 𝑤))
5853, 57syl5bi 232 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
59 iffalse 4067 . . . . . . . . . . . . . . . 16 𝑧𝑎 → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
6059adantr 481 . . . . . . . . . . . . . . 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 2675 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑧) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
62 iffalse 4067 . . . . . . . . . . . . . . . 16 𝑤𝑎 → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
6362adantl 482 . . . . . . . . . . . . . . 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 2675 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑤) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
6561, 64eqeq12d 2636 . . . . . . . . . . . . 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 3189 . . . . . . . . . . . . . . . 16 𝑛 ∈ V
67 vex 3189 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
6866, 67ifex 4128 . . . . . . . . . . . . . . 15 if(𝑧 = 𝑡, 𝑛, 𝑚) ∈ V
6968, 42opth 4905 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 (⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩ → 𝑧 = 𝑤)
7165, 70syl6bi 243 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
7247, 52, 58, 714casesdan 990 . . . . . . . . . . 11 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
7372ralrimivva 2965 . . . . . . . . . 10 ((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → ∀𝑧 ∈ (𝑎𝑏)∀𝑤 ∈ (𝑎𝑏)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
74733ad2ant3 1082 . . . . . . . . 9 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → ∀𝑧 ∈ (𝑎𝑏)∀𝑤 ∈ (𝑎𝑏)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
75 dff13 6466 . . . . . . . . 9 (𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏) ↔ (𝐹:(𝑎𝑏)⟶(𝑎 × 𝑏) ∧ ∀𝑧 ∈ (𝑎𝑏)∀𝑤 ∈ (𝑎𝑏)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
7630, 74, 75sylanbrc 697 . . . . . . . 8 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏))
771, 4unex 6909 . . . . . . . . 9 (𝑎𝑏) ∈ V
781, 4xpex 6915 . . . . . . . . 9 (𝑎 × 𝑏) ∈ V
79 f1dom2g 7917 . . . . . . . . 9 (((𝑎𝑏) ∈ V ∧ (𝑎 × 𝑏) ∈ V ∧ 𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏)) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
8077, 78, 79mp3an12 1411 . . . . . . . 8 (𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
8176, 80syl 17 . . . . . . 7 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
82813expia 1264 . . . . . 6 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏)) → ((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏)))
8382rexlimdvva 3031 . . . . 5 ((𝑚𝑎𝑠𝑏) → (∃𝑛𝑎𝑡𝑏𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏)))
848, 83syl5bir 233 . . . 4 ((𝑚𝑎𝑠𝑏) → ((∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏)))
8584rexlimivv 3029 . . 3 (∃𝑚𝑎𝑠𝑏 (∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
867, 85sylbir 225 . 2 ((∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
873, 6, 86syl2anb 496 1 ((1𝑜𝑎 ∧ 1𝑜𝑏) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cun 3553  ifcif 4058  cop 4154   class class class wbr 4613  cmpt 4673   × cxp 5072  wf 5843  1-1wf1 5844  cfv 5847  1𝑜c1o 7498  cdom 7897  csdm 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1o 7505  df-2o 7506  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902
This theorem is referenced by:  unxpdom  8111
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