Theorem fisseneq 6813

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 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵)

Proof of Theorem fisseneq

Dummy variables 𝑤 𝑥 𝑦 𝑧 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enfii 6761	. . . 4 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
2	1	3adant2 1000	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
3		sseq1 3115	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝑥 ↔ ∅ ⊆ 𝑥))
4		breq1 3927	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑤 ≈ 𝑥 ↔ ∅ ≈ 𝑥))
5	3, 4	anbi12d 464	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) ↔ (∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥)))
6		eqeq1 2144	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 = 𝑥 ↔ ∅ = 𝑥))
7	5, 6	imbi12d 233	. . . . 5 ⊢ (𝑤 = ∅ → (((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ ((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)))
8	7	albidv 1796	. . . 4 ⊢ (𝑤 = ∅ → (∀𝑥((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)))
9		sseq1 3115	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑥))
10		breq1 3927	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ≈ 𝑥 ↔ 𝑦 ≈ 𝑥))
11	9, 10	anbi12d 464	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) ↔ (𝑦 ⊆ 𝑥 ∧ 𝑦 ≈ 𝑥)))
12		eqeq1 2144	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝑥 ↔ 𝑦 = 𝑥))
13	11, 12	imbi12d 233	. . . . 5 ⊢ (𝑤 = 𝑦 → (((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ ((𝑦 ⊆ 𝑥 ∧ 𝑦 ≈ 𝑥) → 𝑦 = 𝑥)))
14	13	albidv 1796	. . . 4 ⊢ (𝑤 = 𝑦 → (∀𝑥((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥((𝑦 ⊆ 𝑥 ∧ 𝑦 ≈ 𝑥) → 𝑦 = 𝑥)))
15		sseq1 3115	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝑥 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑥))
16		breq1 3927	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ≈ 𝑥 ↔ (𝑦 ∪ {𝑧}) ≈ 𝑥))
17	15, 16	anbi12d 464	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)))
18		eqeq1 2144	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 = 𝑥 ↔ (𝑦 ∪ {𝑧}) = 𝑥))
19	17, 18	imbi12d 233	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ (((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
20	19	albidv 1796	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
21		sseq1 3115	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
22		breq1 3927	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ 𝑥 ↔ 𝐴 ≈ 𝑥))
23	21, 22	anbi12d 464	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) ↔ (𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥)))
24		eqeq1 2144	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 = 𝑥 ↔ 𝐴 = 𝑥))
25	23, 24	imbi12d 233	. . . . 5 ⊢ (𝑤 = 𝐴 → (((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ ((𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐴 = 𝑥)))
26	25	albidv 1796	. . . 4 ⊢ (𝑤 = 𝐴 → (∀𝑥((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐴 = 𝑥)))
27		ensym 6668	. . . . . . . 8 ⊢ (∅ ≈ 𝑥 → 𝑥 ≈ ∅)
28		en0 6682	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
29	27, 28	sylib 121	. . . . . . 7 ⊢ (∅ ≈ 𝑥 → 𝑥 = ∅)
30	29	eqcomd 2143	. . . . . 6 ⊢ (∅ ≈ 𝑥 → ∅ = 𝑥)
31	30	adantl 275	. . . . 5 ⊢ ((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)
32	31	ax-gen 1425	. . . 4 ⊢ ∀𝑥((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)
33		sseq2 3116	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑎))
34		breq2 3928	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝑎))
35	33, 34	anbi12d 464	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ((𝑦 ⊆ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ (𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎)))
36		eqeq2 2147	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑎))
37	35, 36	imbi12d 233	. . . . . 6 ⊢ (𝑥 = 𝑎 → (((𝑦 ⊆ 𝑥 ∧ 𝑦 ≈ 𝑥) → 𝑦 = 𝑥) ↔ ((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)))
38	37	cbvalv 1889	. . . . 5 ⊢ (∀𝑥((𝑦 ⊆ 𝑥 ∧ 𝑦 ≈ 𝑥) → 𝑦 = 𝑥) ↔ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎))
39		simplr 519	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎))
40		difun2 3437	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = (𝑦 ∖ {𝑧})
41		difsn 3652	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑦 ∖ {𝑧}) = 𝑦)
42	41	ad3antlr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∖ {𝑧}) = 𝑦)
43	40, 42	syl5eq 2182	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = 𝑦)
44		simprl 520	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) ⊆ 𝑥)
45	44	ssdifd 3207	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑦 ∪ {𝑧}) ∖ {𝑧}) ⊆ (𝑥 ∖ {𝑧}))
46	43, 45	eqsstrrd 3129	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 ⊆ (𝑥 ∖ {𝑧}))
47		simplll 522	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 ∈ Fin)
48		vex 2684	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
49	48	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑧 ∈ V)
50		simpllr 523	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ¬ 𝑧 ∈ 𝑦)
51		unsnfi 6800	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ 𝑦) → (𝑦 ∪ {𝑧}) ∈ Fin)
52	47, 49, 50, 51	syl3anc 1216	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) ∈ Fin)
53		simprr 521	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) ≈ 𝑥)
54		vsnid 3552	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {𝑧}
55		elun2 3239	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
56	54, 55	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
57	56	a1i 9	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
58	44, 57	sseldd 3093	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑧 ∈ 𝑥)
59	52, 53, 57, 58	dif1enen 6767	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑦 ∪ {𝑧}) ∖ {𝑧}) ≈ (𝑥 ∖ {𝑧}))
60	43, 59	eqbrtrrd 3947	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 ≈ (𝑥 ∖ {𝑧}))
61	46, 60	jca 304	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧})))
62		vex 2684	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
63		difexg 4064	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ V → (𝑥 ∖ {𝑧}) ∈ V)
64	62, 63	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑥 ∖ {𝑧}) ∈ V
65		sseq2 3116	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑥 ∖ {𝑧}) → (𝑦 ⊆ 𝑎 ↔ 𝑦 ⊆ (𝑥 ∖ {𝑧})))
66		breq2 3928	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑥 ∖ {𝑧}) → (𝑦 ≈ 𝑎 ↔ 𝑦 ≈ (𝑥 ∖ {𝑧})))
67	65, 66	anbi12d 464	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑥 ∖ {𝑧}) → ((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) ↔ (𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧}))))
68		eqeq2 2147	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑥 ∖ {𝑧}) → (𝑦 = 𝑎 ↔ 𝑦 = (𝑥 ∖ {𝑧})))
69	67, 68	imbi12d 233	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑥 ∖ {𝑧}) → (((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎) ↔ ((𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧})) → 𝑦 = (𝑥 ∖ {𝑧}))))
70	64, 69	spcv 2774	. . . . . . . . . . 11 ⊢ (∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎) → ((𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧})) → 𝑦 = (𝑥 ∖ {𝑧})))
71	39, 61, 70	sylc 62	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 = (𝑥 ∖ {𝑧}))
72	71	uneq1d 3224	. . . . . . . . 9 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) = ((𝑥 ∖ {𝑧}) ∪ {𝑧}))
73	53	ensymd 6670	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑥 ≈ (𝑦 ∪ {𝑧}))
74		enfii 6761	. . . . . . . . . . 11 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝑥 ≈ (𝑦 ∪ {𝑧})) → 𝑥 ∈ Fin)
75	52, 73, 74	syl2anc 408	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑥 ∈ Fin)
76		fidifsnid 6758	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝑧 ∈ 𝑥) → ((𝑥 ∖ {𝑧}) ∪ {𝑧}) = 𝑥)
77	75, 58, 76	syl2anc 408	. . . . . . . . 9 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑥 ∖ {𝑧}) ∪ {𝑧}) = 𝑥)
78	72, 77	eqtrd 2170	. . . . . . . 8 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) = 𝑥)
79	78	ex 114	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) → (((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥))
80	79	alrimiv 1846	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) → ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥))
81	80	ex 114	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎) → ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
82	38, 81	syl5bi 151	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑥((𝑦 ⊆ 𝑥 ∧ 𝑦 ≈ 𝑥) → 𝑦 = 𝑥) → ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
83	8, 14, 20, 26, 32, 82	findcard2s 6777	. . 3 ⊢ (𝐴 ∈ Fin → ∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐴 = 𝑥))
84	2, 83	syl 14	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → ∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐴 = 𝑥))
85		3simpc 980	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → (𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵))
86		sseq2 3116	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐵))
87		breq2 3928	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ 𝐵))
88	86, 87	anbi12d 464	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵)))
89		eqeq2 2147	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵))
90	88, 89	imbi12d 233	. . . 4 ⊢ (𝑥 = 𝐵 → (((𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐴 = 𝑥) ↔ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵)))
91	90	spcgv 2768	. . 3 ⊢ (𝐵 ∈ Fin → (∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐴 = 𝑥) → ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵)))
92	91	3ad2ant1 1002	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → (∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐴 = 𝑥) → ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵)))
93	84, 85, 92	mp2d 47	1 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∧ w3a 962  ∀wal 1329   = wceq 1331   ∈ wcel 1480  Vcvv 2681   ∖ cdif 3063   ∪ cun 3064   ⊆ wss 3066  ∅c0 3358  {csn 3522   class class class wbr 3924   ≈ cen 6625  Fincfn 6627

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1o 6306  df-er 6422  df-en 6628  df-fin 6630

This theorem is referenced by:  phpeqd  6814  f1finf1o  6828  en1eqsn  6829