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Theorem fin17 10370

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 3953	. . . . 5 ⊢ (𝑏 ∈ (On ∖ ω) ↔ (𝑏 ∈ On ∧ ¬ 𝑏 ∈ ω))
2		enfi 9172	. . . . . . . . 9 ⊢ (𝐴 ≈ 𝑏 → (𝐴 ∈ Fin ↔ 𝑏 ∈ Fin))
3		onfin 9212	. . . . . . . . 9 ⊢ (𝑏 ∈ On → (𝑏 ∈ Fin ↔ 𝑏 ∈ ω))
4	2, 3	sylan9bbr 511	. . . . . . . 8 ⊢ ((𝑏 ∈ On ∧ 𝐴 ≈ 𝑏) → (𝐴 ∈ Fin ↔ 𝑏 ∈ ω))
5	4	biimpd 228	. . . . . . 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 452	. . . . 5 ⊢ ((𝑏 ∈ On ∧ ¬ 𝑏 ∈ ω) → (𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin))
8	1, 7	sylbi 216	. . . 4 ⊢ (𝑏 ∈ (On ∖ ω) → (𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin))
9	8	rexlimiv 3147	. . 3 ⊢ (∃𝑏 ∈ (On ∖ ω)𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin)
10	9	con2i 139	. 2 ⊢ (𝐴 ∈ Fin → ¬ ∃𝑏 ∈ (On ∖ ω)𝐴 ≈ 𝑏)
11		isfin7 10277	. 2 ⊢ (𝐴 ∈ Fin → (𝐴 ∈ Fin^VII ↔ ¬ ∃𝑏 ∈ (On ∖ ω)𝐴 ≈ 𝑏))
12	10, 11	mpbird 256	1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ Fin^VII)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2106 ∃wrex 3069 ∖ cdif 3940 class class class wbr 5140 Oncon0 6352 ωcom 7837 ≈ cen 8918 Fincfn 8921 Fin^VIIcfin7 10260

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5291 ax-nul 5298 ax-pr 5419 ax-un 7707

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5141 df-opab 5203 df-mpt 5224 df-tr 5258 df-id 5566 df-eprel 5572 df-po 5580 df-so 5581 df-fr 5623 df-we 5625 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-om 7838 df-1o 8447 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fin7 10267

This theorem is referenced by: fin67 10371 isfin7-2 10372

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