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

Theorem fidifsnen 6860
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 4139 . . . . . 6 (𝑋 ∈ Fin → (𝑋 ∖ {𝐴}) ∈ V)
213ad2ant1 1018 . . . . 5 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ∈ V)
32adantr 276 . . . 4 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ∈ V)
4 enrefg 6754 . . . 4 ((𝑋 ∖ {𝐴}) ∈ V → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐴}))
53, 4syl 14 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐴}))
6 sneq 3600 . . . . 5 (𝐴 = 𝐵 → {𝐴} = {𝐵})
76difeq2d 3251 . . . 4 (𝐴 = 𝐵 → (𝑋 ∖ {𝐴}) = (𝑋 ∖ {𝐵}))
87adantl 277 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) = (𝑋 ∖ {𝐵}))
95, 8breqtrd 4024 . 2 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
102adantr 276 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ∈ V)
11 eqid 2175 . . . 4 (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)) = (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥))
12 iftrue 3537 . . . . . . . 8 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
1312adantl 277 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
14 simpll2 1037 . . . . . . . 8 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝐴𝑋)
1514adantr 276 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → 𝐴𝑋)
1613, 15eqeltrd 2252 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋)
17 simpllr 534 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → ¬ 𝐴 = 𝐵)
1813eqeq1d 2184 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → (if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵𝐴 = 𝐵))
1917, 18mtbird 673 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → ¬ if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵)
2019neneqad 2424 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵)
21 eldifsn 3716 . . . . . 6 (if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}) ↔ (if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋 ∧ if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵))
2216, 20, 21sylanbrc 417 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
23 iffalse 3540 . . . . . . . 8 𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
2423adantl 277 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
25 eldifi 3255 . . . . . . . 8 (𝑥 ∈ (𝑋 ∖ {𝐴}) → 𝑥𝑋)
2625ad2antlr 489 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → 𝑥𝑋)
2724, 26eqeltrd 2252 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋)
28 simpr 110 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 = 𝐵)
2924eqeq1d 2184 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → (if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵𝑥 = 𝐵))
3028, 29mtbird 673 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → ¬ if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵)
3130neneqad 2424 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵)
3227, 31, 21sylanbrc 417 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
33 simpll1 1036 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝑋 ∈ Fin)
3425adantl 277 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝑥𝑋)
35 simpll3 1038 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝐵𝑋)
36 fidceq 6859 . . . . . . 7 ((𝑋 ∈ Fin ∧ 𝑥𝑋𝐵𝑋) → DECID 𝑥 = 𝐵)
3733, 34, 35, 36syl3anc 1238 . . . . . 6 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → DECID 𝑥 = 𝐵)
38 exmiddc 836 . . . . . 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 798 . . . 4 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
41 iftrue 3537 . . . . . . 7 (𝑦 = 𝐴 → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
4241adantl 277 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
43 simpl3 1002 . . . . . . . 8 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵𝑋)
44 simpr 110 . . . . . . . . . 10 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
4544neneqad 2424 . . . . . . . . 9 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐴𝐵)
4645necomd 2431 . . . . . . . 8 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐴)
47 eldifsn 3716 . . . . . . . 8 (𝐵 ∈ (𝑋 ∖ {𝐴}) ↔ (𝐵𝑋𝐵𝐴))
4843, 46, 47sylanbrc 417 . . . . . . 7 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ (𝑋 ∖ {𝐴}))
4948ad2antrr 488 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → 𝐵 ∈ (𝑋 ∖ {𝐴}))
5042, 49eqeltrd 2252 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
51 iffalse 3540 . . . . . . 7 𝑦 = 𝐴 → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
5251adantl 277 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
53 eldifi 3255 . . . . . . . 8 (𝑦 ∈ (𝑋 ∖ {𝐵}) → 𝑦𝑋)
5453ad2antlr 489 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦𝑋)
55 simpr 110 . . . . . . . 8 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → ¬ 𝑦 = 𝐴)
5655neneqad 2424 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦𝐴)
57 eldifsn 3716 . . . . . . 7 (𝑦 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑦𝑋𝑦𝐴))
5854, 56, 57sylanbrc 417 . . . . . 6 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ (𝑋 ∖ {𝐴}))
5952, 58eqeltrd 2252 . . . . 5 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
60 simpll1 1036 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝑋 ∈ Fin)
6153adantl 277 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝑦𝑋)
62 simpll2 1037 . . . . . . 7 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝐴𝑋)
63 fidceq 6859 . . . . . . 7 ((𝑋 ∈ Fin ∧ 𝑦𝑋𝐴𝑋) → DECID 𝑦 = 𝐴)
6460, 61, 62, 63syl3anc 1238 . . . . . 6 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → DECID 𝑦 = 𝐴)
65 exmiddc 836 . . . . . 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 798 . . . 4 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
6812adantl 277 . . . . . . . . . 10 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
6968eqeq2d 2187 . . . . . . . . 9 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝐴))
7069biimpar 297 . . . . . . . 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 110 . . . . . . . . . . 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 2187 . . . . . . . . . . . 12 𝑦 = 𝐴 → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝑦))
7473ad2antlr 489 . . . . . . . . . . 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 147 . . . . . . . . . 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 534 . . . . . . . . . 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 2210 . . . . . . . . 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 531 . . . . . . . . . . . . 13 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑦 ∈ (𝑋 ∖ {𝐵}))
7978ad2antrr 488 . . . . . . . . . . . 12 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ (𝑋 ∖ {𝐵}))
8079eldifbd 3139 . . . . . . . . . . 11 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → ¬ 𝑦 ∈ {𝐵})
8180adantr 276 . . . . . . . . . 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 3606 . . . . . . . . . 10 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
8381, 82sylnib 676 . . . . . . . . 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 620 . . . . . . . 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 115 . . . . . . 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 1036 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑋 ∈ Fin)
8753ad2antll 491 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑦𝑋)
88 simpll2 1037 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝐴𝑋)
8986, 87, 88, 63syl3anc 1238 . . . . . . . . 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 276 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
9271, 85, 91mpjaodan 798 . . . . . 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 2187 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝐵))
9493biimprcd 160 . . . . . . . 8 (𝑥 = 𝐵 → (𝑦 = 𝐴𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
9594adantl 277 . . . . . . 7 (((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = 𝐴𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
9669, 95sylbid 150 . . . . . 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 129 . . . . 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 528 . . . . . . . . 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 277 . . . . . . . . 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 2208 . . . . . . . 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 534 . . . . . . . 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 620 . . . . . . 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 493 . . . . . . . 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 528 . . . . . . . . 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 277 . . . . . . . . 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 2208 . . . . . . . 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 2209 . . . . . . 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 488 . . . . . . 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 798 . . . . . 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 529 . . . . . . . . . . . 12 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑥 ∈ (𝑋 ∖ {𝐴}))
111110eldifbd 3139 . . . . . . . . . . 11 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → ¬ 𝑥 ∈ {𝐴})
112 velsn 3606 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
113111, 112sylnib 676 . . . . . . . . . 10 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → ¬ 𝑥 = 𝐴)
114113ad2antrr 488 . . . . . . . . 9 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → ¬ 𝑥 = 𝐴)
115 simpr 110 . . . . . . . . . . 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 2187 . . . . . . . . . . . 12 𝑥 = 𝐵 → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝑥))
117116ad2antlr 489 . . . . . . . . . . 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 147 . . . . . . . . . 10 ((((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → 𝑦 = 𝑥)
119118eqeq1d 2184 . . . . . . . . 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 673 . . . . . . . 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 2209 . . . . . 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 596 . . . . 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 479 . . . . 5 ((((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵))
12597, 123, 124mpjaodan 798 . . . 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 6066 . . 3 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)):(𝑋 ∖ {𝐴})–1-1-onto→(𝑋 ∖ {𝐵}))
127 f1oeng 6747 . . 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 411 . 2 (((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
129 fidceq 6859 . . 3 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → DECID 𝐴 = 𝐵)
130 exmiddc 836 . . 3 (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
131129, 130syl 14 . 2 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
1329, 128, 131mpjaodan 798 1 ((𝑋 ∈ Fin ∧ 𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  w3a 978   = wceq 1353  wcel 2146  wne 2345  Vcvv 2735  cdif 3124  ifcif 3532  {csn 3589   class class class wbr 3998  cmpt 4059  1-1-ontowf1o 5207  cen 6728  Fincfn 6730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-en 6731  df-fin 6733
This theorem is referenced by:  dif1en  6869
  Copyright terms: Public domain W3C validator