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Theorem fisseneq 6586
Description: A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
Assertion
Ref Expression
fisseneq ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)

Proof of Theorem fisseneq
Dummy variables 𝑤 𝑥 𝑦 𝑧 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enfii 6536 . . . 4 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)
213adant2 960 . . 3 ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → 𝐴 ∈ Fin)
3 sseq1 3036 . . . . . . 7 (𝑤 = ∅ → (𝑤𝑥 ↔ ∅ ⊆ 𝑥))
4 breq1 3823 . . . . . . 7 (𝑤 = ∅ → (𝑤𝑥 ↔ ∅ ≈ 𝑥))
53, 4anbi12d 457 . . . . . 6 (𝑤 = ∅ → ((𝑤𝑥𝑤𝑥) ↔ (∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥)))
6 eqeq1 2091 . . . . . 6 (𝑤 = ∅ → (𝑤 = 𝑥 ↔ ∅ = 𝑥))
75, 6imbi12d 232 . . . . 5 (𝑤 = ∅ → (((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)))
87albidv 1749 . . . 4 (𝑤 = ∅ → (∀𝑥((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)))
9 sseq1 3036 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
10 breq1 3823 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
119, 10anbi12d 457 . . . . . 6 (𝑤 = 𝑦 → ((𝑤𝑥𝑤𝑥) ↔ (𝑦𝑥𝑦𝑥)))
12 eqeq1 2091 . . . . . 6 (𝑤 = 𝑦 → (𝑤 = 𝑥𝑦 = 𝑥))
1311, 12imbi12d 232 . . . . 5 (𝑤 = 𝑦 → (((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ((𝑦𝑥𝑦𝑥) → 𝑦 = 𝑥)))
1413albidv 1749 . . . 4 (𝑤 = 𝑦 → (∀𝑥((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥((𝑦𝑥𝑦𝑥) → 𝑦 = 𝑥)))
15 sseq1 3036 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤𝑥 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑥))
16 breq1 3823 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤𝑥 ↔ (𝑦 ∪ {𝑧}) ≈ 𝑥))
1715, 16anbi12d 457 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤𝑥𝑤𝑥) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)))
18 eqeq1 2091 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 = 𝑥 ↔ (𝑦 ∪ {𝑧}) = 𝑥))
1917, 18imbi12d 232 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ (((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
2019albidv 1749 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
21 sseq1 3036 . . . . . . 7 (𝑤 = 𝐴 → (𝑤𝑥𝐴𝑥))
22 breq1 3823 . . . . . . 7 (𝑤 = 𝐴 → (𝑤𝑥𝐴𝑥))
2321, 22anbi12d 457 . . . . . 6 (𝑤 = 𝐴 → ((𝑤𝑥𝑤𝑥) ↔ (𝐴𝑥𝐴𝑥)))
24 eqeq1 2091 . . . . . 6 (𝑤 = 𝐴 → (𝑤 = 𝑥𝐴 = 𝑥))
2523, 24imbi12d 232 . . . . 5 (𝑤 = 𝐴 → (((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥)))
2625albidv 1749 . . . 4 (𝑤 = 𝐴 → (∀𝑥((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥)))
27 ensym 6444 . . . . . . . 8 (∅ ≈ 𝑥𝑥 ≈ ∅)
28 en0 6458 . . . . . . . 8 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
2927, 28sylib 120 . . . . . . 7 (∅ ≈ 𝑥𝑥 = ∅)
3029eqcomd 2090 . . . . . 6 (∅ ≈ 𝑥 → ∅ = 𝑥)
3130adantl 271 . . . . 5 ((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)
3231ax-gen 1381 . . . 4 𝑥((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)
33 sseq2 3037 . . . . . . . 8 (𝑥 = 𝑎 → (𝑦𝑥𝑦𝑎))
34 breq2 3824 . . . . . . . 8 (𝑥 = 𝑎 → (𝑦𝑥𝑦𝑎))
3533, 34anbi12d 457 . . . . . . 7 (𝑥 = 𝑎 → ((𝑦𝑥𝑦𝑥) ↔ (𝑦𝑎𝑦𝑎)))
36 eqeq2 2094 . . . . . . 7 (𝑥 = 𝑎 → (𝑦 = 𝑥𝑦 = 𝑎))
3735, 36imbi12d 232 . . . . . 6 (𝑥 = 𝑎 → (((𝑦𝑥𝑦𝑥) → 𝑦 = 𝑥) ↔ ((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)))
3837cbvalv 1839 . . . . 5 (∀𝑥((𝑦𝑥𝑦𝑥) → 𝑦 = 𝑥) ↔ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎))
39 simplr 497 . . . . . . . . . . 11 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎))
40 difun2 3349 . . . . . . . . . . . . . 14 ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = (𝑦 ∖ {𝑧})
41 difsn 3557 . . . . . . . . . . . . . . 15 𝑧𝑦 → (𝑦 ∖ {𝑧}) = 𝑦)
4241ad3antlr 477 . . . . . . . . . . . . . 14 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∖ {𝑧}) = 𝑦)
4340, 42syl5eq 2129 . . . . . . . . . . . . 13 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = 𝑦)
44 simprl 498 . . . . . . . . . . . . . 14 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) ⊆ 𝑥)
4544ssdifd 3125 . . . . . . . . . . . . 13 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑦 ∪ {𝑧}) ∖ {𝑧}) ⊆ (𝑥 ∖ {𝑧}))
4643, 45eqsstr3d 3050 . . . . . . . . . . . 12 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 ⊆ (𝑥 ∖ {𝑧}))
47 simplll 500 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 ∈ Fin)
48 vex 2618 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
4948a1i 9 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑧 ∈ V)
50 simpllr 501 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ¬ 𝑧𝑦)
51 unsnfi 6575 . . . . . . . . . . . . . . 15 ((𝑦 ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧𝑦) → (𝑦 ∪ {𝑧}) ∈ Fin)
5247, 49, 50, 51syl3anc 1172 . . . . . . . . . . . . . 14 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) ∈ Fin)
53 simprr 499 . . . . . . . . . . . . . 14 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) ≈ 𝑥)
54 vsnid 3459 . . . . . . . . . . . . . . . 16 𝑧 ∈ {𝑧}
55 elun2 3157 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
5654, 55ax-mp 7 . . . . . . . . . . . . . . 15 𝑧 ∈ (𝑦 ∪ {𝑧})
5756a1i 9 . . . . . . . . . . . . . 14 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
5844, 57sseldd 3015 . . . . . . . . . . . . . 14 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑧𝑥)
5952, 53, 57, 58dif1enen 6542 . . . . . . . . . . . . 13 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑦 ∪ {𝑧}) ∖ {𝑧}) ≈ (𝑥 ∖ {𝑧}))
6043, 59eqbrtrrd 3842 . . . . . . . . . . . 12 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 ≈ (𝑥 ∖ {𝑧}))
6146, 60jca 300 . . . . . . . . . . 11 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧})))
62 vex 2618 . . . . . . . . . . . . 13 𝑥 ∈ V
63 difexg 3954 . . . . . . . . . . . . 13 (𝑥 ∈ V → (𝑥 ∖ {𝑧}) ∈ V)
6462, 63ax-mp 7 . . . . . . . . . . . 12 (𝑥 ∖ {𝑧}) ∈ V
65 sseq2 3037 . . . . . . . . . . . . . 14 (𝑎 = (𝑥 ∖ {𝑧}) → (𝑦𝑎𝑦 ⊆ (𝑥 ∖ {𝑧})))
66 breq2 3824 . . . . . . . . . . . . . 14 (𝑎 = (𝑥 ∖ {𝑧}) → (𝑦𝑎𝑦 ≈ (𝑥 ∖ {𝑧})))
6765, 66anbi12d 457 . . . . . . . . . . . . 13 (𝑎 = (𝑥 ∖ {𝑧}) → ((𝑦𝑎𝑦𝑎) ↔ (𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧}))))
68 eqeq2 2094 . . . . . . . . . . . . 13 (𝑎 = (𝑥 ∖ {𝑧}) → (𝑦 = 𝑎𝑦 = (𝑥 ∖ {𝑧})))
6967, 68imbi12d 232 . . . . . . . . . . . 12 (𝑎 = (𝑥 ∖ {𝑧}) → (((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎) ↔ ((𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧})) → 𝑦 = (𝑥 ∖ {𝑧}))))
7064, 69spcv 2705 . . . . . . . . . . 11 (∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎) → ((𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧})) → 𝑦 = (𝑥 ∖ {𝑧})))
7139, 61, 70sylc 61 . . . . . . . . . 10 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 = (𝑥 ∖ {𝑧}))
7271uneq1d 3142 . . . . . . . . 9 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) = ((𝑥 ∖ {𝑧}) ∪ {𝑧}))
7353ensymd 6446 . . . . . . . . . . 11 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑥 ≈ (𝑦 ∪ {𝑧}))
74 enfii 6536 . . . . . . . . . . 11 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝑥 ≈ (𝑦 ∪ {𝑧})) → 𝑥 ∈ Fin)
7552, 73, 74syl2anc 403 . . . . . . . . . 10 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑥 ∈ Fin)
76 fidifsnid 6533 . . . . . . . . . 10 ((𝑥 ∈ Fin ∧ 𝑧𝑥) → ((𝑥 ∖ {𝑧}) ∪ {𝑧}) = 𝑥)
7775, 58, 76syl2anc 403 . . . . . . . . 9 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑥 ∖ {𝑧}) ∪ {𝑧}) = 𝑥)
7872, 77eqtrd 2117 . . . . . . . 8 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) = 𝑥)
7978ex 113 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) → (((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥))
8079alrimiv 1799 . . . . . 6 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) → ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥))
8180ex 113 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎) → ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
8238, 81syl5bi 150 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑥((𝑦𝑥𝑦𝑥) → 𝑦 = 𝑥) → ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
838, 14, 20, 26, 32, 82findcard2s 6552 . . 3 (𝐴 ∈ Fin → ∀𝑥((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥))
842, 83syl 14 . 2 ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → ∀𝑥((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥))
85 3simpc 940 . 2 ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → (𝐴𝐵𝐴𝐵))
86 sseq2 3037 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
87 breq2 3824 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
8886, 87anbi12d 457 . . . . 5 (𝑥 = 𝐵 → ((𝐴𝑥𝐴𝑥) ↔ (𝐴𝐵𝐴𝐵)))
89 eqeq2 2094 . . . . 5 (𝑥 = 𝐵 → (𝐴 = 𝑥𝐴 = 𝐵))
9088, 89imbi12d 232 . . . 4 (𝑥 = 𝐵 → (((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥) ↔ ((𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)))
9190spcgv 2699 . . 3 (𝐵 ∈ Fin → (∀𝑥((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥) → ((𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)))
92913ad2ant1 962 . 2 ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → (∀𝑥((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥) → ((𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)))
9384, 85, 92mp2d 46 1 ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  w3a 922  wal 1285   = wceq 1287  wcel 1436  Vcvv 2615  cdif 2985  cun 2986  wss 2988  c0 3275  {csn 3431   class class class wbr 3820  cen 6401  Fincfn 6403
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3928  ax-sep 3931  ax-nul 3939  ax-pow 3983  ax-pr 4009  ax-un 4233  ax-setind 4325  ax-iinf 4375
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3911  df-id 4093  df-iord 4166  df-on 4168  df-suc 4171  df-iom 4378  df-xp 4416  df-rel 4417  df-cnv 4418  df-co 4419  df-dm 4420  df-rn 4421  df-res 4422  df-ima 4423  df-iota 4943  df-fun 4980  df-fn 4981  df-f 4982  df-f1 4983  df-fo 4984  df-f1o 4985  df-fv 4986  df-1o 6129  df-er 6238  df-en 6404  df-fin 6406
This theorem is referenced by:  f1finf1o  6599  en1eqsn  6600
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