ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fidifsnen GIF version

Theorem fidifsnen 6294
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 3895 . . . . . 6 (𝑋 ∈ Fin → (𝑋 ∖ {𝐴}) ∈ V)
213ad2ant1 925 . . . . 5 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ∈ V)
32adantr 261 . . . 4 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ∈ V)
4 enrefg 6207 . . . 4 ((𝑋 ∖ {𝐴}) ∈ V → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐴}))
53, 4syl 14 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐴}))
6 sneq 3383 . . . . 5 (𝐴 = 𝐵 → {𝐴} = {𝐵})
76difeq2d 3059 . . . 4 (𝐴 = 𝐵 → (𝑋 ∖ {𝐴}) = (𝑋 ∖ {𝐵}))
87adantl 262 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) = (𝑋 ∖ {𝐵}))
95, 8breqtrd 3785 . 2 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
102adantr 261 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ∈ V)
11 eqid 2040 . . . 4 (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)) = (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥))
12 iftrue 3333 . . . . . . . 8 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
1312adantl 262 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
14 simpll2 944 . . . . . . . 8 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝐴𝑋)
1514adantr 261 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → 𝐴𝑋)
1613, 15eqeltrd 2114 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋)
17 simpllr 486 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → ¬ 𝐴 = 𝐵)
1813eqeq1d 2048 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → (if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵𝐴 = 𝐵))
1917, 18mtbird 598 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → ¬ if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵)
2019neneqad 2281 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵)
21 eldifsn 3492 . . . . . 6 (if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}) ↔ (if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋 ∧ if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵))
2216, 20, 21sylanbrc 394 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
23 iffalse 3336 . . . . . . . 8 𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
2423adantl 262 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
25 eldifi 3063 . . . . . . . 8 (𝑥 ∈ (𝑋 ∖ {𝐴}) → 𝑥𝑋)
2625ad2antlr 458 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → 𝑥𝑋)
2724, 26eqeltrd 2114 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋)
28 simpr 103 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 = 𝐵)
2924eqeq1d 2048 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → (if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵𝑥 = 𝐵))
3028, 29mtbird 598 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → ¬ if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵)
3130neneqad 2281 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵)
3227, 31, 21sylanbrc 394 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
33 simpll1 943 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝑋 ∈ Fin)
3425adantl 262 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝑥𝑋)
35 simpll3 945 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝐵𝑋)
36 fidceq 6293 . . . . . . 7 ((𝑋 ∈ Fin ∧ 𝑥𝑋𝐵𝑋) → DECID 𝑥 = 𝐵)
3733, 34, 35, 36syl3anc 1135 . . . . . 6 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → DECID 𝑥 = 𝐵)
38 exmiddc 744 . . . . . 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 711 . . . 4 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
41 iftrue 3333 . . . . . . 7 (𝑦 = 𝐴 → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
4241adantl 262 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
43 simpl3 909 . . . . . . . 8 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵𝑋)
44 simpr 103 . . . . . . . . . 10 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
4544neneqad 2281 . . . . . . . . 9 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐴𝐵)
4645necomd 2288 . . . . . . . 8 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐴)
47 eldifsn 3492 . . . . . . . 8 (𝐵 ∈ (𝑋 ∖ {𝐴}) ↔ (𝐵𝑋𝐵𝐴))
4843, 46, 47sylanbrc 394 . . . . . . 7 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ (𝑋 ∖ {𝐴}))
4948ad2antrr 457 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → 𝐵 ∈ (𝑋 ∖ {𝐴}))
5042, 49eqeltrd 2114 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
51 iffalse 3336 . . . . . . 7 𝑦 = 𝐴 → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
5251adantl 262 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
53 eldifi 3063 . . . . . . . 8 (𝑦 ∈ (𝑋 ∖ {𝐵}) → 𝑦𝑋)
5453ad2antlr 458 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦𝑋)
55 simpr 103 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → ¬ 𝑦 = 𝐴)
5655neneqad 2281 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦𝐴)
57 eldifsn 3492 . . . . . . 7 (𝑦 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑦𝑋𝑦𝐴))
5854, 56, 57sylanbrc 394 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ (𝑋 ∖ {𝐴}))
5952, 58eqeltrd 2114 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
60 simpll1 943 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝑋 ∈ Fin)
6153adantl 262 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝑦𝑋)
62 simpll2 944 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝐴𝑋)
63 fidceq 6293 . . . . . . 7 ((𝑋 ∈ Fin ∧ 𝑦𝑋𝐴𝑋) → DECID 𝑦 = 𝐴)
6460, 61, 62, 63syl3anc 1135 . . . . . 6 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → DECID 𝑦 = 𝐴)
65 exmiddc 744 . . . . . 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 711 . . . 4 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
6812adantl 262 . . . . . . . . . 10 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
6968eqeq2d 2051 . . . . . . . . 9 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝐴))
7069biimpar 281 . . . . . . . 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 103 . . . . . . . . . . 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 2051 . . . . . . . . . . . 12 𝑦 = 𝐴 → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝑦))
7473ad2antlr 458 . . . . . . . . . . 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 135 . . . . . . . . . 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 486 . . . . . . . . . 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 2074 . . . . . . . . 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 484 . . . . . . . . . . . . 13 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑦 ∈ (𝑋 ∖ {𝐵}))
7978ad2antrr 457 . . . . . . . . . . . 12 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ (𝑋 ∖ {𝐵}))
8079eldifbd 2927 . . . . . . . . . . 11 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → ¬ 𝑦 ∈ {𝐵})
8180adantr 261 . . . . . . . . . 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 3389 . . . . . . . . . 10 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
8381, 82sylnib 601 . . . . . . . . 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 550 . . . . . . . 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 108 . . . . . . 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 943 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑋 ∈ Fin)
8753ad2antll 460 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑦𝑋)
88 simpll2 944 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝐴𝑋)
8986, 87, 88, 63syl3anc 1135 . . . . . . . . 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 261 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
9271, 85, 91mpjaodan 711 . . . . . 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 2051 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝐵))
9493biimprcd 149 . . . . . . . 8 (𝑥 = 𝐵 → (𝑦 = 𝐴𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
9594adantl 262 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = 𝐴𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
9669, 95sylbid 139 . . . . . 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 120 . . . . 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 482 . . . . . . . . 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 262 . . . . . . . . 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 2072 . . . . . . . 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 486 . . . . . . . 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 550 . . . . . . 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 462 . . . . . . . 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 482 . . . . . . . . 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 262 . . . . . . . . 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 2072 . . . . . . . 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 2073 . . . . . . 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 457 . . . . . . 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 711 . . . . . 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 483 . . . . . . . . . . . 12 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑥 ∈ (𝑋 ∖ {𝐴}))
111110eldifbd 2927 . . . . . . . . . . 11 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → ¬ 𝑥 ∈ {𝐴})
112 velsn 3389 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
113111, 112sylnib 601 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → ¬ 𝑥 = 𝐴)
114113ad2antrr 457 . . . . . . . . 9 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → ¬ 𝑥 = 𝐴)
115 simpr 103 . . . . . . . . . . 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 2051 . . . . . . . . . . . 12 𝑥 = 𝐵 → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝑥))
117116ad2antlr 458 . . . . . . . . . . 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 135 . . . . . . . . . 10 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → 𝑦 = 𝑥)
119118eqeq1d 2048 . . . . . . . . 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 598 . . . . . . . 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 2073 . . . . . 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 528 . . . . 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 448 . . . . 5 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵))
12597, 123, 124mpjaodan 711 . . . 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 5668 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)):(𝑋 ∖ {𝐴})–1-1-onto→(𝑋 ∖ {𝐵}))
127 f1oeng 6200 . . 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 391 . 2 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
129 fidceq 6293 . . 3 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → DECID 𝐴 = 𝐵)
130 exmiddc 744 . . 3 (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
131129, 130syl 14 . 2 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
1329, 128, 131mpjaodan 711 1 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  wo 629  DECID wdc 742  w3a 885   = wceq 1243  wcel 1393  wne 2204  Vcvv 2554  cdif 2911  ifcif 3328  {csn 3372   class class class wbr 3761  cmpt 3815  1-1-ontowf1o 4864  cen 6182  Fincfn 6184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-if 3329  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-id 4027  df-iord 4075  df-on 4077  df-suc 4080  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-en 6185  df-fin 6187
This theorem is referenced by:  dif1en  6300
  Copyright terms: Public domain W3C validator