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Theorem fidifsnen 6516
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 3945 . . . . . 6 (𝑋 ∈ Fin → (𝑋 ∖ {𝐴}) ∈ V)
213ad2ant1 960 . . . . 5 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ∈ V)
32adantr 270 . . . 4 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ∈ V)
4 enrefg 6411 . . . 4 ((𝑋 ∖ {𝐴}) ∈ V → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐴}))
53, 4syl 14 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐴}))
6 sneq 3433 . . . . 5 (𝐴 = 𝐵 → {𝐴} = {𝐵})
76difeq2d 3102 . . . 4 (𝐴 = 𝐵 → (𝑋 ∖ {𝐴}) = (𝑋 ∖ {𝐵}))
87adantl 271 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) = (𝑋 ∖ {𝐵}))
95, 8breqtrd 3835 . 2 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
102adantr 270 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ∈ V)
11 eqid 2083 . . . 4 (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)) = (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥))
12 iftrue 3378 . . . . . . . 8 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
1312adantl 271 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
14 simpll2 979 . . . . . . . 8 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝐴𝑋)
1514adantr 270 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → 𝐴𝑋)
1613, 15eqeltrd 2159 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋)
17 simpllr 501 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → ¬ 𝐴 = 𝐵)
1813eqeq1d 2091 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → (if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵𝐴 = 𝐵))
1917, 18mtbird 631 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → ¬ if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵)
2019neneqad 2328 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵)
21 eldifsn 3541 . . . . . 6 (if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}) ↔ (if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋 ∧ if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵))
2216, 20, 21sylanbrc 408 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
23 iffalse 3381 . . . . . . . 8 𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
2423adantl 271 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
25 eldifi 3106 . . . . . . . 8 (𝑥 ∈ (𝑋 ∖ {𝐴}) → 𝑥𝑋)
2625ad2antlr 473 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → 𝑥𝑋)
2724, 26eqeltrd 2159 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋)
28 simpr 108 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 = 𝐵)
2924eqeq1d 2091 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → (if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵𝑥 = 𝐵))
3028, 29mtbird 631 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → ¬ if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵)
3130neneqad 2328 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵)
3227, 31, 21sylanbrc 408 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
33 simpll1 978 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝑋 ∈ Fin)
3425adantl 271 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝑥𝑋)
35 simpll3 980 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝐵𝑋)
36 fidceq 6515 . . . . . . 7 ((𝑋 ∈ Fin ∧ 𝑥𝑋𝐵𝑋) → DECID 𝑥 = 𝐵)
3733, 34, 35, 36syl3anc 1170 . . . . . 6 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → DECID 𝑥 = 𝐵)
38 exmiddc 778 . . . . . 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 745 . . . 4 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
41 iftrue 3378 . . . . . . 7 (𝑦 = 𝐴 → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
4241adantl 271 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
43 simpl3 944 . . . . . . . 8 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵𝑋)
44 simpr 108 . . . . . . . . . 10 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
4544neneqad 2328 . . . . . . . . 9 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐴𝐵)
4645necomd 2335 . . . . . . . 8 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐴)
47 eldifsn 3541 . . . . . . . 8 (𝐵 ∈ (𝑋 ∖ {𝐴}) ↔ (𝐵𝑋𝐵𝐴))
4843, 46, 47sylanbrc 408 . . . . . . 7 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ (𝑋 ∖ {𝐴}))
4948ad2antrr 472 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → 𝐵 ∈ (𝑋 ∖ {𝐴}))
5042, 49eqeltrd 2159 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
51 iffalse 3381 . . . . . . 7 𝑦 = 𝐴 → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
5251adantl 271 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
53 eldifi 3106 . . . . . . . 8 (𝑦 ∈ (𝑋 ∖ {𝐵}) → 𝑦𝑋)
5453ad2antlr 473 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦𝑋)
55 simpr 108 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → ¬ 𝑦 = 𝐴)
5655neneqad 2328 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦𝐴)
57 eldifsn 3541 . . . . . . 7 (𝑦 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑦𝑋𝑦𝐴))
5854, 56, 57sylanbrc 408 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ (𝑋 ∖ {𝐴}))
5952, 58eqeltrd 2159 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
60 simpll1 978 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝑋 ∈ Fin)
6153adantl 271 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝑦𝑋)
62 simpll2 979 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝐴𝑋)
63 fidceq 6515 . . . . . . 7 ((𝑋 ∈ Fin ∧ 𝑦𝑋𝐴𝑋) → DECID 𝑦 = 𝐴)
6460, 61, 62, 63syl3anc 1170 . . . . . 6 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → DECID 𝑦 = 𝐴)
65 exmiddc 778 . . . . . 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 745 . . . 4 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
6812adantl 271 . . . . . . . . . 10 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
6968eqeq2d 2094 . . . . . . . . 9 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝐴))
7069biimpar 291 . . . . . . . 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 108 . . . . . . . . . . 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 2094 . . . . . . . . . . . 12 𝑦 = 𝐴 → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝑦))
7473ad2antlr 473 . . . . . . . . . . 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 145 . . . . . . . . . 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 501 . . . . . . . . . 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 2117 . . . . . . . . 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 499 . . . . . . . . . . . . 13 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑦 ∈ (𝑋 ∖ {𝐵}))
7978ad2antrr 472 . . . . . . . . . . . 12 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ (𝑋 ∖ {𝐵}))
8079eldifbd 2996 . . . . . . . . . . 11 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → ¬ 𝑦 ∈ {𝐵})
8180adantr 270 . . . . . . . . . 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 3439 . . . . . . . . . 10 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
8381, 82sylnib 634 . . . . . . . . 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 583 . . . . . . . 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 113 . . . . . . 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 978 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑋 ∈ Fin)
8753ad2antll 475 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑦𝑋)
88 simpll2 979 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝐴𝑋)
8986, 87, 88, 63syl3anc 1170 . . . . . . . . 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 270 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
9271, 85, 91mpjaodan 745 . . . . . 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 2094 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝐵))
9493biimprcd 158 . . . . . . . 8 (𝑥 = 𝐵 → (𝑦 = 𝐴𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
9594adantl 271 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = 𝐴𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
9669, 95sylbid 148 . . . . . 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 127 . . . . 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 497 . . . . . . . . 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 271 . . . . . . . . 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 2115 . . . . . . . 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 501 . . . . . . . 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 583 . . . . . . 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 477 . . . . . . . 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 497 . . . . . . . . 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 271 . . . . . . . . 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 2115 . . . . . . . 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 2116 . . . . . . 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 472 . . . . . . 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 745 . . . . . 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 498 . . . . . . . . . . . 12 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑥 ∈ (𝑋 ∖ {𝐴}))
111110eldifbd 2996 . . . . . . . . . . 11 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → ¬ 𝑥 ∈ {𝐴})
112 velsn 3439 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
113111, 112sylnib 634 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → ¬ 𝑥 = 𝐴)
114113ad2antrr 472 . . . . . . . . 9 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → ¬ 𝑥 = 𝐴)
115 simpr 108 . . . . . . . . . . 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 2094 . . . . . . . . . . . 12 𝑥 = 𝐵 → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝑥))
117116ad2antlr 473 . . . . . . . . . . 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 145 . . . . . . . . . 10 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → 𝑦 = 𝑥)
119118eqeq1d 2091 . . . . . . . . 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 631 . . . . . . . 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 2116 . . . . . 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 561 . . . . 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 463 . . . . 5 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵))
12597, 123, 124mpjaodan 745 . . . 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 5784 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)):(𝑋 ∖ {𝐴})–1-1-onto→(𝑋 ∖ {𝐵}))
127 f1oeng 6404 . . 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 403 . 2 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
129 fidceq 6515 . . 3 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → DECID 𝐴 = 𝐵)
130 exmiddc 778 . . 3 (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
131129, 130syl 14 . 2 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
1329, 128, 131mpjaodan 745 1 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 776  w3a 920   = wceq 1285  wcel 1434  wne 2249  Vcvv 2612  cdif 2981  ifcif 3373  {csn 3422   class class class wbr 3811  cmpt 3865  1-1-ontowf1o 4968  cen 6385  Fincfn 6387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-en 6388  df-fin 6390
This theorem is referenced by:  dif1en  6525
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