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Theorem fidifsnen 6771
Description: All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)
Assertion
Ref Expression
fidifsnen ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))

Proof of Theorem fidifsnen
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difexg 4076 . . . . . 6 (𝑋 ∈ Fin → (𝑋 ∖ {𝐴}) ∈ V)
213ad2ant1 1003 . . . . 5 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ∈ V)
32adantr 274 . . . 4 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ∈ V)
4 enrefg 6665 . . . 4 ((𝑋 ∖ {𝐴}) ∈ V → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐴}))
53, 4syl 14 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐴}))
6 sneq 3542 . . . . 5 (𝐴 = 𝐵 → {𝐴} = {𝐵})
76difeq2d 3198 . . . 4 (𝐴 = 𝐵 → (𝑋 ∖ {𝐴}) = (𝑋 ∖ {𝐵}))
87adantl 275 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) = (𝑋 ∖ {𝐵}))
95, 8breqtrd 3961 . 2 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
102adantr 274 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ∈ V)
11 eqid 2140 . . . 4 (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)) = (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥))
12 iftrue 3483 . . . . . . . 8 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
1312adantl 275 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
14 simpll2 1022 . . . . . . . 8 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝐴𝑋)
1514adantr 274 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → 𝐴𝑋)
1613, 15eqeltrd 2217 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋)
17 simpllr 524 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → ¬ 𝐴 = 𝐵)
1813eqeq1d 2149 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → (if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵𝐴 = 𝐵))
1917, 18mtbird 663 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → ¬ if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵)
2019neneqad 2388 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵)
21 eldifsn 3657 . . . . . 6 (if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}) ↔ (if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋 ∧ if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵))
2216, 20, 21sylanbrc 414 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
23 iffalse 3486 . . . . . . . 8 𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
2423adantl 275 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
25 eldifi 3202 . . . . . . . 8 (𝑥 ∈ (𝑋 ∖ {𝐴}) → 𝑥𝑋)
2625ad2antlr 481 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → 𝑥𝑋)
2724, 26eqeltrd 2217 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋)
28 simpr 109 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 = 𝐵)
2924eqeq1d 2149 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → (if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵𝑥 = 𝐵))
3028, 29mtbird 663 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → ¬ if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵)
3130neneqad 2388 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵)
3227, 31, 21sylanbrc 414 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
33 simpll1 1021 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝑋 ∈ Fin)
3425adantl 275 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝑥𝑋)
35 simpll3 1023 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝐵𝑋)
36 fidceq 6770 . . . . . . 7 ((𝑋 ∈ Fin ∧ 𝑥𝑋𝐵𝑋) → DECID 𝑥 = 𝐵)
3733, 34, 35, 36syl3anc 1217 . . . . . 6 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → DECID 𝑥 = 𝐵)
38 exmiddc 822 . . . . . 6 (DECID 𝑥 = 𝐵 → (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵))
3937, 38syl 14 . . . . 5 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵))
4022, 32, 39mpjaodan 788 . . . 4 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
41 iftrue 3483 . . . . . . 7 (𝑦 = 𝐴 → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
4241adantl 275 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
43 simpl3 987 . . . . . . . 8 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵𝑋)
44 simpr 109 . . . . . . . . . 10 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
4544neneqad 2388 . . . . . . . . 9 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐴𝐵)
4645necomd 2395 . . . . . . . 8 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐴)
47 eldifsn 3657 . . . . . . . 8 (𝐵 ∈ (𝑋 ∖ {𝐴}) ↔ (𝐵𝑋𝐵𝐴))
4843, 46, 47sylanbrc 414 . . . . . . 7 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ (𝑋 ∖ {𝐴}))
4948ad2antrr 480 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → 𝐵 ∈ (𝑋 ∖ {𝐴}))
5042, 49eqeltrd 2217 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
51 iffalse 3486 . . . . . . 7 𝑦 = 𝐴 → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
5251adantl 275 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
53 eldifi 3202 . . . . . . . 8 (𝑦 ∈ (𝑋 ∖ {𝐵}) → 𝑦𝑋)
5453ad2antlr 481 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦𝑋)
55 simpr 109 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → ¬ 𝑦 = 𝐴)
5655neneqad 2388 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦𝐴)
57 eldifsn 3657 . . . . . . 7 (𝑦 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑦𝑋𝑦𝐴))
5854, 56, 57sylanbrc 414 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ (𝑋 ∖ {𝐴}))
5952, 58eqeltrd 2217 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
60 simpll1 1021 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝑋 ∈ Fin)
6153adantl 275 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝑦𝑋)
62 simpll2 1022 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝐴𝑋)
63 fidceq 6770 . . . . . . 7 ((𝑋 ∈ Fin ∧ 𝑦𝑋𝐴𝑋) → DECID 𝑦 = 𝐴)
6460, 61, 62, 63syl3anc 1217 . . . . . 6 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → DECID 𝑦 = 𝐴)
65 exmiddc 822 . . . . . 6 (DECID 𝑦 = 𝐴 → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
6664, 65syl 14 . . . . 5 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
6750, 59, 66mpjaodan 788 . . . 4 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
6812adantl 275 . . . . . . . . . 10 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
6968eqeq2d 2152 . . . . . . . . 9 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝐴))
7069biimpar 295 . . . . . . . 8 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ 𝑦 = 𝐴) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥))
7170a1d 22 . . . . . . 7 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ 𝑦 = 𝐴) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)))
72 simpr 109 . . . . . . . . . . 11 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦))
7351eqeq2d 2152 . . . . . . . . . . . 12 𝑦 = 𝐴 → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝑦))
7473ad2antlr 481 . . . . . . . . . . 11 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝑦))
7572, 74mpbid 146 . . . . . . . . . 10 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → 𝑥 = 𝑦)
76 simpllr 524 . . . . . . . . . 10 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → 𝑥 = 𝐵)
7775, 76eqtr3d 2175 . . . . . . . . 9 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → 𝑦 = 𝐵)
78 simprr 522 . . . . . . . . . . . . 13 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑦 ∈ (𝑋 ∖ {𝐵}))
7978ad2antrr 480 . . . . . . . . . . . 12 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ (𝑋 ∖ {𝐵}))
8079eldifbd 3087 . . . . . . . . . . 11 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → ¬ 𝑦 ∈ {𝐵})
8180adantr 274 . . . . . . . . . 10 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → ¬ 𝑦 ∈ {𝐵})
82 velsn 3548 . . . . . . . . . 10 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
8381, 82sylnib 666 . . . . . . . . 9 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → ¬ 𝑦 = 𝐵)
8477, 83pm2.21dd 610 . . . . . . . 8 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥))
8584ex 114 . . . . . . 7 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)))
86 simpll1 1021 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑋 ∈ Fin)
8753ad2antll 483 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑦𝑋)
88 simpll2 1022 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝐴𝑋)
8986, 87, 88, 63syl3anc 1217 . . . . . . . . 9 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → DECID 𝑦 = 𝐴)
9089, 65syl 14 . . . . . . . 8 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
9190adantr 274 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
9271, 85, 91mpjaodan 788 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)))
9341eqeq2d 2152 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝐵))
9493biimprcd 159 . . . . . . . 8 (𝑥 = 𝐵 → (𝑦 = 𝐴𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
9594adantl 275 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = 𝐴𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
9669, 95sylbid 149 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) → 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
9792, 96impbid 128 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)))
98 simplr 520 . . . . . . . . 9 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ 𝑦 = 𝐴) → 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦))
9941adantl 275 . . . . . . . . 9 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
10098, 99eqtrd 2173 . . . . . . . 8 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ 𝑦 = 𝐴) → 𝑥 = 𝐵)
101 simpllr 524 . . . . . . . 8 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ 𝑦 = 𝐴) → ¬ 𝑥 = 𝐵)
102100, 101pm2.21dd 610 . . . . . . 7 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ 𝑦 = 𝐴) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥))
10323ad3antlr 485 . . . . . . . 8 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ ¬ 𝑦 = 𝐴) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
104 simplr 520 . . . . . . . . 9 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ ¬ 𝑦 = 𝐴) → 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦))
10551adantl 275 . . . . . . . . 9 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
106104, 105eqtrd 2173 . . . . . . . 8 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ ¬ 𝑦 = 𝐴) → 𝑥 = 𝑦)
107103, 106eqtr2d 2174 . . . . . . 7 (((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ ¬ 𝑦 = 𝐴) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥))
10890ad2antrr 480 . . . . . . 7 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
109102, 107, 108mpjaodan 788 . . . . . 6 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥))
110 simprl 521 . . . . . . . . . . . 12 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑥 ∈ (𝑋 ∖ {𝐴}))
111110eldifbd 3087 . . . . . . . . . . 11 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → ¬ 𝑥 ∈ {𝐴})
112 velsn 3548 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
113111, 112sylnib 666 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → ¬ 𝑥 = 𝐴)
114113ad2antrr 480 . . . . . . . . 9 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → ¬ 𝑥 = 𝐴)
115 simpr 109 . . . . . . . . . . 11 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥))
11623eqeq2d 2152 . . . . . . . . . . . 12 𝑥 = 𝐵 → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝑥))
117116ad2antlr 481 . . . . . . . . . . 11 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝑥))
118115, 117mpbid 146 . . . . . . . . . 10 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → 𝑦 = 𝑥)
119118eqeq1d 2149 . . . . . . . . 9 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → (𝑦 = 𝐴𝑥 = 𝐴))
120114, 119mtbird 663 . . . . . . . 8 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → ¬ 𝑦 = 𝐴)
121120, 51syl 14 . . . . . . 7 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
122121, 118eqtr2d 2174 . . . . . 6 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦))
123109, 122impbida 586 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)))
12439adantrr 471 . . . . 5 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵))
12597, 123, 124mpjaodan 788 . . . 4 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)))
12611, 40, 67, 125f1o2d 5982 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)):(𝑋 ∖ {𝐴})–1-1-onto→(𝑋 ∖ {𝐵}))
127 f1oeng 6658 . . 3 (((𝑋 ∖ {𝐴}) ∈ V ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)):(𝑋 ∖ {𝐴})–1-1-onto→(𝑋 ∖ {𝐵})) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
12810, 126, 127syl2anc 409 . 2 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
129 fidceq 6770 . . 3 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → DECID 𝐴 = 𝐵)
130 exmiddc 822 . . 3 (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
131129, 130syl 14 . 2 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
1329, 128, 131mpjaodan 788 1 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  DECID wdc 820  w3a 963   = wceq 1332  wcel 1481  wne 2309  Vcvv 2689  cdif 3072  ifcif 3478  {csn 3531   class class class wbr 3936  cmpt 3996  1-1-ontowf1o 5129  cen 6639  Fincfn 6641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-en 6642  df-fin 6644
This theorem is referenced by:  dif1en  6780
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