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Theorem fin17 10434

Description: Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.)

**Assertion**
Ref	Expression
fin17	⊢ (𝐴 ∈ Fin → 𝐴 ∈ Fin^VII)

Proof of Theorem fin17 Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3961	. . . . 5 ⊢ (𝑏 ∈ (On ∖ ω) ↔ (𝑏 ∈ On ∧ ¬ 𝑏 ∈ ω))
2		enfi 9227	. . . . . . . . 9 ⊢ (𝐴 ≈ 𝑏 → (𝐴 ∈ Fin ↔ 𝑏 ∈ Fin))
3		onfin 9267	. . . . . . . . 9 ⊢ (𝑏 ∈ On → (𝑏 ∈ Fin ↔ 𝑏 ∈ ω))
4	2, 3	sylan9bbr 510	. . . . . . . 8 ⊢ ((𝑏 ∈ On ∧ 𝐴 ≈ 𝑏) → (𝐴 ∈ Fin ↔ 𝑏 ∈ ω))
5	4	biimpd 229	. . . . . . 7 ⊢ ((𝑏 ∈ On ∧ 𝐴 ≈ 𝑏) → (𝐴 ∈ Fin → 𝑏 ∈ ω))
6	5	con3d 152	. . . . . 6 ⊢ ((𝑏 ∈ On ∧ 𝐴 ≈ 𝑏) → (¬ 𝑏 ∈ ω → ¬ 𝐴 ∈ Fin))
7	6	impancom 451	. . . . 5 ⊢ ((𝑏 ∈ On ∧ ¬ 𝑏 ∈ ω) → (𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin))
8	1, 7	sylbi 217	. . . 4 ⊢ (𝑏 ∈ (On ∖ ω) → (𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin))
9	8	rexlimiv 3148	. . 3 ⊢ (∃𝑏 ∈ (On ∖ ω)𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin)
10	9	con2i 139	. 2 ⊢ (𝐴 ∈ Fin → ¬ ∃𝑏 ∈ (On ∖ ω)𝐴 ≈ 𝑏)
11		isfin7 10341	. 2 ⊢ (𝐴 ∈ Fin → (𝐴 ∈ Fin^VII ↔ ¬ ∃𝑏 ∈ (On ∖ ω)𝐴 ≈ 𝑏))
12	10, 11	mpbird 257	1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ Fin^VII)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2108 ∃wrex 3070 ∖ cdif 3948 class class class wbr 5143 Oncon0 6384 ωcom 7887 ≈ cen 8982 Fincfn 8985 Fin^VIIcfin7 10324

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fin7 10331

This theorem is referenced by: fin67 10435 isfin7-2 10436

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