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Theorem fidifsnen 6362
 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 3926 . . . . . 6 (𝑋 ∈ Fin → (𝑋 ∖ {𝐴}) ∈ V)
213ad2ant1 936 . . . . 5 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ∈ V)
32adantr 265 . . . 4 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ∈ V)
4 enrefg 6275 . . . 4 ((𝑋 ∖ {𝐴}) ∈ V → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐴}))
53, 4syl 14 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐴}))
6 sneq 3414 . . . . 5 (𝐴 = 𝐵 → {𝐴} = {𝐵})
76difeq2d 3090 . . . 4 (𝐴 = 𝐵 → (𝑋 ∖ {𝐴}) = (𝑋 ∖ {𝐵}))
87adantl 266 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) = (𝑋 ∖ {𝐵}))
95, 8breqtrd 3816 . 2 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
102adantr 265 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ∈ V)
11 eqid 2056 . . . 4 (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)) = (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥))
12 iftrue 3364 . . . . . . . 8 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
1312adantl 266 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
14 simpll2 955 . . . . . . . 8 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝐴𝑋)
1514adantr 265 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → 𝐴𝑋)
1613, 15eqeltrd 2130 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋)
17 simpllr 494 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → ¬ 𝐴 = 𝐵)
1813eqeq1d 2064 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → (if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵𝐴 = 𝐵))
1917, 18mtbird 608 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → ¬ if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵)
2019neneqad 2299 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵)
21 eldifsn 3523 . . . . . 6 (if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}) ↔ (if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋 ∧ if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵))
2216, 20, 21sylanbrc 402 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
23 iffalse 3367 . . . . . . . 8 𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
2423adantl 266 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
25 eldifi 3094 . . . . . . . 8 (𝑥 ∈ (𝑋 ∖ {𝐴}) → 𝑥𝑋)
2625ad2antlr 466 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → 𝑥𝑋)
2724, 26eqeltrd 2130 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋)
28 simpr 107 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 = 𝐵)
2924eqeq1d 2064 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → (if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵𝑥 = 𝐵))
3028, 29mtbird 608 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → ¬ if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵)
3130neneqad 2299 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵)
3227, 31, 21sylanbrc 402 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
33 simpll1 954 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝑋 ∈ Fin)
3425adantl 266 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝑥𝑋)
35 simpll3 956 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝐵𝑋)
36 fidceq 6361 . . . . . . 7 ((𝑋 ∈ Fin ∧ 𝑥𝑋𝐵𝑋) → DECID 𝑥 = 𝐵)
3733, 34, 35, 36syl3anc 1146 . . . . . 6 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → DECID 𝑥 = 𝐵)
38 exmiddc 755 . . . . . 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 722 . . . 4 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
41 iftrue 3364 . . . . . . 7 (𝑦 = 𝐴 → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
4241adantl 266 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
43 simpl3 920 . . . . . . . 8 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵𝑋)
44 simpr 107 . . . . . . . . . 10 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
4544neneqad 2299 . . . . . . . . 9 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐴𝐵)
4645necomd 2306 . . . . . . . 8 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐴)
47 eldifsn 3523 . . . . . . . 8 (𝐵 ∈ (𝑋 ∖ {𝐴}) ↔ (𝐵𝑋𝐵𝐴))
4843, 46, 47sylanbrc 402 . . . . . . 7 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ (𝑋 ∖ {𝐴}))
4948ad2antrr 465 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → 𝐵 ∈ (𝑋 ∖ {𝐴}))
5042, 49eqeltrd 2130 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
51 iffalse 3367 . . . . . . 7 𝑦 = 𝐴 → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
5251adantl 266 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
53 eldifi 3094 . . . . . . . 8 (𝑦 ∈ (𝑋 ∖ {𝐵}) → 𝑦𝑋)
5453ad2antlr 466 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦𝑋)
55 simpr 107 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → ¬ 𝑦 = 𝐴)
5655neneqad 2299 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦𝐴)
57 eldifsn 3523 . . . . . . 7 (𝑦 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑦𝑋𝑦𝐴))
5854, 56, 57sylanbrc 402 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ (𝑋 ∖ {𝐴}))
5952, 58eqeltrd 2130 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
60 simpll1 954 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝑋 ∈ Fin)
6153adantl 266 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝑦𝑋)
62 simpll2 955 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝐴𝑋)
63 fidceq 6361 . . . . . . 7 ((𝑋 ∈ Fin ∧ 𝑦𝑋𝐴𝑋) → DECID 𝑦 = 𝐴)
6460, 61, 62, 63syl3anc 1146 . . . . . 6 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → DECID 𝑦 = 𝐴)
65 exmiddc 755 . . . . . 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 722 . . . 4 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
6812adantl 266 . . . . . . . . . 10 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
6968eqeq2d 2067 . . . . . . . . 9 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝐴))
7069biimpar 285 . . . . . . . 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 107 . . . . . . . . . . 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 2067 . . . . . . . . . . . 12 𝑦 = 𝐴 → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝑦))
7473ad2antlr 466 . . . . . . . . . . 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 139 . . . . . . . . . 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 494 . . . . . . . . . 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 2090 . . . . . . . . 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 492 . . . . . . . . . . . . 13 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑦 ∈ (𝑋 ∖ {𝐵}))
7978ad2antrr 465 . . . . . . . . . . . 12 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ (𝑋 ∖ {𝐵}))
8079eldifbd 2958 . . . . . . . . . . 11 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → ¬ 𝑦 ∈ {𝐵})
8180adantr 265 . . . . . . . . . 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 3420 . . . . . . . . . 10 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
8381, 82sylnib 611 . . . . . . . . 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 560 . . . . . . . 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 112 . . . . . . 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 954 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑋 ∈ Fin)
8753ad2antll 468 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑦𝑋)
88 simpll2 955 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝐴𝑋)
8986, 87, 88, 63syl3anc 1146 . . . . . . . . 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 265 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
9271, 85, 91mpjaodan 722 . . . . . 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 2067 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝐵))
9493biimprcd 153 . . . . . . . 8 (𝑥 = 𝐵 → (𝑦 = 𝐴𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
9594adantl 266 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = 𝐴𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
9669, 95sylbid 143 . . . . . 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 124 . . . . 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 490 . . . . . . . . 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 266 . . . . . . . . 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 2088 . . . . . . . 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 494 . . . . . . . 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 560 . . . . . . 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 470 . . . . . . . 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 490 . . . . . . . . 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 266 . . . . . . . . 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 2088 . . . . . . . 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 2089 . . . . . . 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 465 . . . . . . 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 722 . . . . . 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 491 . . . . . . . . . . . 12 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑥 ∈ (𝑋 ∖ {𝐴}))
111110eldifbd 2958 . . . . . . . . . . 11 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → ¬ 𝑥 ∈ {𝐴})
112 velsn 3420 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
113111, 112sylnib 611 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → ¬ 𝑥 = 𝐴)
114113ad2antrr 465 . . . . . . . . 9 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → ¬ 𝑥 = 𝐴)
115 simpr 107 . . . . . . . . . . 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 2067 . . . . . . . . . . . 12 𝑥 = 𝐵 → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝑥))
117116ad2antlr 466 . . . . . . . . . . 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 139 . . . . . . . . . 10 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → 𝑦 = 𝑥)
119118eqeq1d 2064 . . . . . . . . 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 608 . . . . . . . 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 2089 . . . . . 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 538 . . . . 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 456 . . . . 5 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵))
12597, 123, 124mpjaodan 722 . . . 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 5733 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)):(𝑋 ∖ {𝐴})–1-1-onto→(𝑋 ∖ {𝐵}))
127 f1oeng 6268 . . 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 397 . 2 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
129 fidceq 6361 . . 3 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → DECID 𝐴 = 𝐵)
130 exmiddc 755 . . 3 (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
131129, 130syl 14 . 2 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
1329, 128, 131mpjaodan 722 1 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ↔ wb 102   ∨ wo 639  DECID wdc 753   ∧ w3a 896   = wceq 1259   ∈ wcel 1409   ≠ wne 2220  Vcvv 2574   ∖ cdif 2942  ifcif 3359  {csn 3403   class class class wbr 3792   ↦ cmpt 3846  –1-1-onto→wf1o 4929   ≈ cen 6250  Fincfn 6252 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339 This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-if 3360  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-en 6253  df-fin 6255 This theorem is referenced by:  dif1en  6368
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