fin17 - Metamath Proof Explorer

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Theorem fin17 10482

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 4051	. . . . 5 ⊢ (𝑏 ∈ (On ∖ ω) ↔ (𝑏 ∈ On ∧ ¬ 𝑏 ∈ ω))
2		enfi 9213	. . . . . . . . 9 ⊢ (𝐴 ≈ 𝑏 → (𝐴 ∈ Fin ↔ 𝑏 ∈ Fin))
3		onfin 9241	. . . . . . . . 9 ⊢ (𝑏 ∈ On → (𝑏 ∈ Fin ↔ 𝑏 ∈ ω))
4	2, 3	sylan9bbr 520	. . . . . . . 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 457	. . . . 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 10389	. 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∈ wcel 2146 ∃wrex 3089 ∖ cdif 4047 class class class wbr 5105 Oncon0 6368 ωcom 7884 ≈ cen 8981 Fincfn 8984 Fin^VIIcfin7 10372

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5251 ax-nul 5262 ax-pr 5394 ax-un 7758

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 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 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-om 7885 df-1o 8487 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fin7 10379

This theorem is used by: fin67 10483 isfin7-2 10484

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