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Theorem fisseneq 6828
 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 6776 . . . 4 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)
213adant2 1001 . . 3 ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → 𝐴 ∈ Fin)
3 sseq1 3125 . . . . . . 7 (𝑤 = ∅ → (𝑤𝑥 ↔ ∅ ⊆ 𝑥))
4 breq1 3940 . . . . . . 7 (𝑤 = ∅ → (𝑤𝑥 ↔ ∅ ≈ 𝑥))
53, 4anbi12d 465 . . . . . 6 (𝑤 = ∅ → ((𝑤𝑥𝑤𝑥) ↔ (∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥)))
6 eqeq1 2147 . . . . . 6 (𝑤 = ∅ → (𝑤 = 𝑥 ↔ ∅ = 𝑥))
75, 6imbi12d 233 . . . . 5 (𝑤 = ∅ → (((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)))
87albidv 1797 . . . 4 (𝑤 = ∅ → (∀𝑥((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)))
9 sseq1 3125 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
10 breq1 3940 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
119, 10anbi12d 465 . . . . . 6 (𝑤 = 𝑦 → ((𝑤𝑥𝑤𝑥) ↔ (𝑦𝑥𝑦𝑥)))
12 eqeq1 2147 . . . . . 6 (𝑤 = 𝑦 → (𝑤 = 𝑥𝑦 = 𝑥))
1311, 12imbi12d 233 . . . . 5 (𝑤 = 𝑦 → (((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ((𝑦𝑥𝑦𝑥) → 𝑦 = 𝑥)))
1413albidv 1797 . . . 4 (𝑤 = 𝑦 → (∀𝑥((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥((𝑦𝑥𝑦𝑥) → 𝑦 = 𝑥)))
15 sseq1 3125 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤𝑥 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑥))
16 breq1 3940 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤𝑥 ↔ (𝑦 ∪ {𝑧}) ≈ 𝑥))
1715, 16anbi12d 465 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤𝑥𝑤𝑥) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)))
18 eqeq1 2147 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 = 𝑥 ↔ (𝑦 ∪ {𝑧}) = 𝑥))
1917, 18imbi12d 233 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ (((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
2019albidv 1797 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
21 sseq1 3125 . . . . . . 7 (𝑤 = 𝐴 → (𝑤𝑥𝐴𝑥))
22 breq1 3940 . . . . . . 7 (𝑤 = 𝐴 → (𝑤𝑥𝐴𝑥))
2321, 22anbi12d 465 . . . . . 6 (𝑤 = 𝐴 → ((𝑤𝑥𝑤𝑥) ↔ (𝐴𝑥𝐴𝑥)))
24 eqeq1 2147 . . . . . 6 (𝑤 = 𝐴 → (𝑤 = 𝑥𝐴 = 𝑥))
2523, 24imbi12d 233 . . . . 5 (𝑤 = 𝐴 → (((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥)))
2625albidv 1797 . . . 4 (𝑤 = 𝐴 → (∀𝑥((𝑤𝑥𝑤𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥)))
27 ensym 6683 . . . . . . . 8 (∅ ≈ 𝑥𝑥 ≈ ∅)
28 en0 6697 . . . . . . . 8 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
2927, 28sylib 121 . . . . . . 7 (∅ ≈ 𝑥𝑥 = ∅)
3029eqcomd 2146 . . . . . 6 (∅ ≈ 𝑥 → ∅ = 𝑥)
3130adantl 275 . . . . 5 ((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)
3231ax-gen 1426 . . . 4 𝑥((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)
33 sseq2 3126 . . . . . . . 8 (𝑥 = 𝑎 → (𝑦𝑥𝑦𝑎))
34 breq2 3941 . . . . . . . 8 (𝑥 = 𝑎 → (𝑦𝑥𝑦𝑎))
3533, 34anbi12d 465 . . . . . . 7 (𝑥 = 𝑎 → ((𝑦𝑥𝑦𝑥) ↔ (𝑦𝑎𝑦𝑎)))
36 eqeq2 2150 . . . . . . 7 (𝑥 = 𝑎 → (𝑦 = 𝑥𝑦 = 𝑎))
3735, 36imbi12d 233 . . . . . 6 (𝑥 = 𝑎 → (((𝑦𝑥𝑦𝑥) → 𝑦 = 𝑥) ↔ ((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)))
3837cbvalv 1890 . . . . 5 (∀𝑥((𝑦𝑥𝑦𝑥) → 𝑦 = 𝑥) ↔ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎))
39 simplr 520 . . . . . . . . . . 11 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎))
40 difun2 3447 . . . . . . . . . . . . . 14 ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = (𝑦 ∖ {𝑧})
41 difsn 3665 . . . . . . . . . . . . . . 15 𝑧𝑦 → (𝑦 ∖ {𝑧}) = 𝑦)
4241ad3antlr 485 . . . . . . . . . . . . . 14 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∖ {𝑧}) = 𝑦)
4340, 42syl5eq 2185 . . . . . . . . . . . . 13 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = 𝑦)
44 simprl 521 . . . . . . . . . . . . . 14 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) ⊆ 𝑥)
4544ssdifd 3217 . . . . . . . . . . . . 13 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑦 ∪ {𝑧}) ∖ {𝑧}) ⊆ (𝑥 ∖ {𝑧}))
4643, 45eqsstrrd 3139 . . . . . . . . . . . 12 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 ⊆ (𝑥 ∖ {𝑧}))
47 simplll 523 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 ∈ Fin)
48 vex 2692 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
4948a1i 9 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑧 ∈ V)
50 simpllr 524 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ¬ 𝑧𝑦)
51 unsnfi 6815 . . . . . . . . . . . . . . 15 ((𝑦 ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧𝑦) → (𝑦 ∪ {𝑧}) ∈ Fin)
5247, 49, 50, 51syl3anc 1217 . . . . . . . . . . . . . 14 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) ∈ Fin)
53 simprr 522 . . . . . . . . . . . . . 14 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) ≈ 𝑥)
54 vsnid 3564 . . . . . . . . . . . . . . . 16 𝑧 ∈ {𝑧}
55 elun2 3249 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
5654, 55ax-mp 5 . . . . . . . . . . . . . . 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 3103 . . . . . . . . . . . . . 14 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑧𝑥)
5952, 53, 57, 58dif1enen 6782 . . . . . . . . . . . . 13 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑦 ∪ {𝑧}) ∖ {𝑧}) ≈ (𝑥 ∖ {𝑧}))
6043, 59eqbrtrrd 3960 . . . . . . . . . . . 12 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 ≈ (𝑥 ∖ {𝑧}))
6146, 60jca 304 . . . . . . . . . . 11 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧})))
62 vex 2692 . . . . . . . . . . . . 13 𝑥 ∈ V
63 difexg 4077 . . . . . . . . . . . . 13 (𝑥 ∈ V → (𝑥 ∖ {𝑧}) ∈ V)
6462, 63ax-mp 5 . . . . . . . . . . . 12 (𝑥 ∖ {𝑧}) ∈ V
65 sseq2 3126 . . . . . . . . . . . . . 14 (𝑎 = (𝑥 ∖ {𝑧}) → (𝑦𝑎𝑦 ⊆ (𝑥 ∖ {𝑧})))
66 breq2 3941 . . . . . . . . . . . . . 14 (𝑎 = (𝑥 ∖ {𝑧}) → (𝑦𝑎𝑦 ≈ (𝑥 ∖ {𝑧})))
6765, 66anbi12d 465 . . . . . . . . . . . . 13 (𝑎 = (𝑥 ∖ {𝑧}) → ((𝑦𝑎𝑦𝑎) ↔ (𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧}))))
68 eqeq2 2150 . . . . . . . . . . . . 13 (𝑎 = (𝑥 ∖ {𝑧}) → (𝑦 = 𝑎𝑦 = (𝑥 ∖ {𝑧})))
6967, 68imbi12d 233 . . . . . . . . . . . 12 (𝑎 = (𝑥 ∖ {𝑧}) → (((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎) ↔ ((𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧})) → 𝑦 = (𝑥 ∖ {𝑧}))))
7064, 69spcv 2783 . . . . . . . . . . 11 (∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎) → ((𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧})) → 𝑦 = (𝑥 ∖ {𝑧})))
7139, 61, 70sylc 62 . . . . . . . . . 10 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 = (𝑥 ∖ {𝑧}))
7271uneq1d 3234 . . . . . . . . 9 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) = ((𝑥 ∖ {𝑧}) ∪ {𝑧}))
7353ensymd 6685 . . . . . . . . . . 11 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑥 ≈ (𝑦 ∪ {𝑧}))
74 enfii 6776 . . . . . . . . . . 11 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝑥 ≈ (𝑦 ∪ {𝑧})) → 𝑥 ∈ Fin)
7552, 73, 74syl2anc 409 . . . . . . . . . 10 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑥 ∈ Fin)
76 fidifsnid 6773 . . . . . . . . . 10 ((𝑥 ∈ Fin ∧ 𝑧𝑥) → ((𝑥 ∖ {𝑧}) ∪ {𝑧}) = 𝑥)
7775, 58, 76syl2anc 409 . . . . . . . . 9 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑥 ∖ {𝑧}) ∪ {𝑧}) = 𝑥)
7872, 77eqtrd 2173 . . . . . . . 8 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) = 𝑥)
7978ex 114 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) → (((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥))
8079alrimiv 1847 . . . . . 6 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ ∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎)) → ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥))
8180ex 114 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑎((𝑦𝑎𝑦𝑎) → 𝑦 = 𝑎) → ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
8238, 81syl5bi 151 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑥((𝑦𝑥𝑦𝑥) → 𝑦 = 𝑥) → ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
838, 14, 20, 26, 32, 82findcard2s 6792 . . 3 (𝐴 ∈ Fin → ∀𝑥((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥))
842, 83syl 14 . 2 ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → ∀𝑥((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥))
85 3simpc 981 . 2 ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → (𝐴𝐵𝐴𝐵))
86 sseq2 3126 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
87 breq2 3941 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
8886, 87anbi12d 465 . . . . 5 (𝑥 = 𝐵 → ((𝐴𝑥𝐴𝑥) ↔ (𝐴𝐵𝐴𝐵)))
89 eqeq2 2150 . . . . 5 (𝑥 = 𝐵 → (𝐴 = 𝑥𝐴 = 𝐵))
9088, 89imbi12d 233 . . . 4 (𝑥 = 𝐵 → (((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥) ↔ ((𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)))
9190spcgv 2776 . . 3 (𝐵 ∈ Fin → (∀𝑥((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥) → ((𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)))
92913ad2ant1 1003 . 2 ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → (∀𝑥((𝐴𝑥𝐴𝑥) → 𝐴 = 𝑥) → ((𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)))
9384, 85, 92mp2d 47 1 ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∧ w3a 963  ∀wal 1330   = wceq 1332   ∈ wcel 1481  Vcvv 2689   ∖ cdif 3073   ∪ cun 3074   ⊆ wss 3076  ∅c0 3368  {csn 3532   class class class wbr 3937   ≈ cen 6640  Fincfn 6642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1o 6321  df-er 6437  df-en 6643  df-fin 6645 This theorem is referenced by:  phpeqd  6829  f1finf1o  6843  en1eqsn  6844
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