Theorem fisseneq 6909

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 6852	. . . 4 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
2	1	3adant2 1011	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
3		sseq1 3170	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝑥 ↔ ∅ ⊆ 𝑥))
4		breq1 3992	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑤 ≈ 𝑥 ↔ ∅ ≈ 𝑥))
5	3, 4	anbi12d 470	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) ↔ (∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥)))
6		eqeq1 2177	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 = 𝑥 ↔ ∅ = 𝑥))
7	5, 6	imbi12d 233	. . . . 5 ⊢ (𝑤 = ∅ → (((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ ((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)))
8	7	albidv 1817	. . . 4 ⊢ (𝑤 = ∅ → (∀𝑥((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)))
9		sseq1 3170	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑥))
10		breq1 3992	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ≈ 𝑥 ↔ 𝑦 ≈ 𝑥))
11	9, 10	anbi12d 470	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) ↔ (𝑦 ⊆ 𝑥 ∧ 𝑦 ≈ 𝑥)))
12		eqeq1 2177	. . . . . 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 1817	. . . 4 ⊢ (𝑤 = 𝑦 → (∀𝑥((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥((𝑦 ⊆ 𝑥 ∧ 𝑦 ≈ 𝑥) → 𝑦 = 𝑥)))
15		sseq1 3170	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝑥 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑥))
16		breq1 3992	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ≈ 𝑥 ↔ (𝑦 ∪ {𝑧}) ≈ 𝑥))
17	15, 16	anbi12d 470	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)))
18		eqeq1 2177	. . . . . 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 1817	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥(((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥) → (𝑦 ∪ {𝑧}) = 𝑥)))
21		sseq1 3170	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
22		breq1 3992	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ 𝑥 ↔ 𝐴 ≈ 𝑥))
23	21, 22	anbi12d 470	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) ↔ (𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥)))
24		eqeq1 2177	. . . . . 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 1817	. . . 4 ⊢ (𝑤 = 𝐴 → (∀𝑥((𝑤 ⊆ 𝑥 ∧ 𝑤 ≈ 𝑥) → 𝑤 = 𝑥) ↔ ∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐴 = 𝑥)))
27		ensym 6759	. . . . . . . 8 ⊢ (∅ ≈ 𝑥 → 𝑥 ≈ ∅)
28		en0 6773	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
29	27, 28	sylib 121	. . . . . . 7 ⊢ (∅ ≈ 𝑥 → 𝑥 = ∅)
30	29	eqcomd 2176	. . . . . 6 ⊢ (∅ ≈ 𝑥 → ∅ = 𝑥)
31	30	adantl 275	. . . . 5 ⊢ ((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)
32	31	ax-gen 1442	. . . 4 ⊢ ∀𝑥((∅ ⊆ 𝑥 ∧ ∅ ≈ 𝑥) → ∅ = 𝑥)
33		sseq2 3171	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑎))
34		breq2 3993	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝑎))
35	33, 34	anbi12d 470	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ((𝑦 ⊆ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ (𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎)))
36		eqeq2 2180	. . . . . . 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 1910	. . . . 5 ⊢ (∀𝑥((𝑦 ⊆ 𝑥 ∧ 𝑦 ≈ 𝑥) → 𝑦 = 𝑥) ↔ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎))
39		simplr 525	. . . . . . . . . . 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 3494	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = (𝑦 ∖ {𝑧})
41		difsn 3717	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑦 ∖ {𝑧}) = 𝑦)
42	41	ad3antlr 490	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∖ {𝑧}) = 𝑦)
43	40, 42	eqtrid 2215	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = 𝑦)
44		simprl 526	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) ⊆ 𝑥)
45	44	ssdifd 3263	. . . . . . . . . . . . 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 3184	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 ⊆ (𝑥 ∖ {𝑧}))
47		simplll 528	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑦 ∈ Fin)
48		vex 2733	. . . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → ¬ 𝑧 ∈ 𝑦)
51		unsnfi 6896	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ 𝑦) → (𝑦 ∪ {𝑧}) ∈ Fin)
52	47, 49, 50, 51	syl3anc 1233	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) ∈ Fin)
53		simprr 527	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → (𝑦 ∪ {𝑧}) ≈ 𝑥)
54		vsnid 3615	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ {𝑧}
55		elun2 3295	. . . . . . . . . . . . . . . 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 3148	. . . . . . . . . . . . . 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 6858	. . . . . . . . . . . . 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 4013	. . . . . . . . . . . 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 2733	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
63		difexg 4130	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ V → (𝑥 ∖ {𝑧}) ∈ V)
64	62, 63	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑥 ∖ {𝑧}) ∈ V
65		sseq2 3171	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑥 ∖ {𝑧}) → (𝑦 ⊆ 𝑎 ↔ 𝑦 ⊆ (𝑥 ∖ {𝑧})))
66		breq2 3993	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑥 ∖ {𝑧}) → (𝑦 ≈ 𝑎 ↔ 𝑦 ≈ (𝑥 ∖ {𝑧})))
67	65, 66	anbi12d 470	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑥 ∖ {𝑧}) → ((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) ↔ (𝑦 ⊆ (𝑥 ∖ {𝑧}) ∧ 𝑦 ≈ (𝑥 ∖ {𝑧}))))
68		eqeq2 2180	. . . . . . . . . . . . 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 2824	. . . . . . . . . . 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 3280	. . . . . . . . 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 6761	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑥 ≈ (𝑦 ∪ {𝑧}))
74		enfii 6852	. . . . . . . . . . 11 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝑥 ≈ (𝑦 ∪ {𝑧})) → 𝑥 ∈ Fin)
75	52, 73, 74	syl2anc 409	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑎((𝑦 ⊆ 𝑎 ∧ 𝑦 ≈ 𝑎) → 𝑦 = 𝑎)) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑥 ∧ (𝑦 ∪ {𝑧}) ≈ 𝑥)) → 𝑥 ∈ Fin)
76		fidifsnid 6849	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝑧 ∈ 𝑥) → ((𝑥 ∖ {𝑧}) ∪ {𝑧}) = 𝑥)
77	75, 58, 76	syl2anc 409	. . . . . . . . 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 2203	. . . . . . . 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 1867	. . . . . 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 6868	. . 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 991	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → (𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵))
86		sseq2 3171	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐵))
87		breq2 3993	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ 𝐵))
88	86, 87	anbi12d 470	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵)))
89		eqeq2 2180	. . . . 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 2817	. . 3 ⊢ (𝐵 ∈ Fin → (∀𝑥((𝐴 ⊆ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐴 = 𝑥) → ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵)))
92	91	3ad2ant1 1013	. 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 973  ∀wal 1346   = wceq 1348   ∈ wcel 2141  Vcvv 2730   ∖ cdif 3118   ∪ cun 3119   ⊆ wss 3121  ∅c0 3414  {csn 3583   class class class wbr 3989   ≈ cen 6716  Fincfn 6718

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1o 6395  df-er 6513  df-en 6719  df-fin 6721

This theorem is referenced by:  phpeqd  6910  f1finf1o  6924  en1eqsn  6925