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Theorem fin17 10304

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 3911	. . . . 5 ⊢ (𝑏 ∈ (On ∖ ω) ↔ (𝑏 ∈ On ∧ ¬ 𝑏 ∈ ω))
2		enfi 9111	. . . . . . . . 9 ⊢ (𝐴 ≈ 𝑏 → (𝐴 ∈ Fin ↔ 𝑏 ∈ Fin))
3		onfin 9139	. . . . . . . . 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 3130	. . 3 ⊢ (∃𝑏 ∈ (On ∖ ω)𝐴 ≈ 𝑏 → ¬ 𝐴 ∈ Fin)
10	9	con2i 139	. 2 ⊢ (𝐴 ∈ Fin → ¬ ∃𝑏 ∈ (On ∖ ω)𝐴 ≈ 𝑏)
11		isfin7 10211	. 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 2113 ∃wrex 3060 ∖ cdif 3898 class class class wbr 5098 Oncon0 6317 ωcom 7808 ≈ cen 8880 Fincfn 8883 Fin^VIIcfin7 10194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-om 7809 df-1o 8397 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fin7 10201

This theorem is referenced by: fin67 10305 isfin7-2 10306

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