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Theorem fin17 10373

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 3915	. . . . 5 ⊢ (𝑏 ∈ (On ∖ ω) ↔ (𝑏 ∈ On ∧ ¬ 𝑏 ∈ ω))
2		enfi 9167	. . . . . . . . 9 ⊢ (𝐴 ≈ 𝑏 → (𝐴 ∈ Fin ↔ 𝑏 ∈ Fin))
3		onfin 9195	. . . . . . . . 9 ⊢ (𝑏 ∈ On → (𝑏 ∈ Fin ↔ 𝑏 ∈ ω))
4	2, 3	sylan9bbr 519	. . . . . . . 8 ⊢ ((𝑏 ∈ On ∧ 𝐴 ≈ 𝑏) → (𝐴 ∈ Fin ↔ 𝑏 ∈ ω))
5	4	biimpd 232	. . . . . . 7 ⊢ ((𝑏 ∈ On ∧ 𝐴 ≈ 𝑏) → (𝐴 ∈ Fin → 𝑏 ∈ ω))
6	5	con3d 153	. . . . . 6 ⊢ ((𝑏 ∈ On ∧ 𝐴 ≈ 𝑏) → (¬ 𝑏 ∈ ω → ¬ 𝐴 ∈ Fin))
7	6	impancom 456	. . . . 5 ⊢ ((𝑏 ∈ On ∧ ¬ 𝑏 ∈ ω) → (𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin))
8	1, 7	sylbi 220	. . . 4 ⊢ (𝑏 ∈ (On ∖ ω) → (𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin))
9	8	rexlimiv 3159	. . 3 ⊢ (∃𝑏 ∈ (On ∖ ω)𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin)
10	9	con2i 140	. 2 ⊢ (𝐴 ∈ Fin → ¬ ∃𝑏 ∈ (On ∖ ω)𝐴 ≈ 𝑏)
11		isfin7 10280	. 2 ⊢ (𝐴 ∈ Fin → (𝐴 ∈ Fin^VII ↔ ¬ ∃𝑏 ∈ (On ∖ ω)𝐴 ≈ 𝑏))
12	10, 11	mpbird 260	1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ Fin^VII)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∈ wcel 2143 ∃wrex 3089 ∖ cdif 3902 class class class wbr 5109 Oncon0 6360 ωcom 7858 ≈ cen 8936 Fincfn 8939 Fin^VIIcfin7 10263

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-om 7859 df-1o 8449 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fin7 10270

This theorem is referenced by: fin67 10374 isfin7-2 10375

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