Theorem isfin7-2 10484

Description: A set is VII-finite iff it is non-well-orderable or finite. (Contributed by Mario Carneiro, 17-May-2015.)

Assertion

Ref	Expression
isfin7-2	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin)))

Proof of Theorem isfin7-2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfin7 10389	. . . 4 ⊢ (𝐴 ∈ FinVII → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
2	1	ibi 270	. . 3 ⊢ (𝐴 ∈ FinVII → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥)
3		isnum2 10036	. . . . 5 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
4		ensym 9041	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ≈ 𝑥)
5		simprl 783	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝑥 ∈ On)
6		enfi 9213	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≈ 𝑥 → (𝐴 ∈ Fin ↔ 𝑥 ∈ Fin))
7		onfin 9241	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ On → (𝑥 ∈ Fin ↔ 𝑥 ∈ ω))
8	6, 7	sylan9bbr 520	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → (𝐴 ∈ Fin ↔ 𝑥 ∈ ω))
9	8	biimprd 251	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → (𝑥 ∈ ω → 𝐴 ∈ Fin))
10	9	con3d 153	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → (¬ 𝐴 ∈ Fin → ¬ 𝑥 ∈ ω))
11	10	impcom 413	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → ¬ 𝑥 ∈ ω)
12	5, 11	eldifd 4052	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝑥 ∈ (On ∖ ω))
13		simprr 785	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝐴 ≈ 𝑥)
14	12, 13	jca 521	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥))
15	4, 14	sylanr2 696	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝑥 ≈ 𝐴)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥))
16	15	ex 418	. . . . . . 7 ⊢ (¬ 𝐴 ∈ Fin → ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥)))
17	16	reximdv2 3175	. . . . . 6 ⊢ (¬ 𝐴 ∈ Fin → (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
18	17	com12 33	. . . . 5 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
19	3, 18	sylbi 220	. . . 4 ⊢ (𝐴 ∈ dom card → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
20	19	con1d 146	. . 3 ⊢ (𝐴 ∈ dom card → (¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥 → 𝐴 ∈ Fin))
21	2, 20	syl5com 32	. 2 ⊢ (𝐴 ∈ FinVII → (𝐴 ∈ dom card → 𝐴 ∈ Fin))
22		eldifi 4069	. . . . . . 7 ⊢ (𝑥 ∈ (On ∖ ω) → 𝑥 ∈ On)
23		ensym 9041	. . . . . . 7 ⊢ (𝐴 ≈ 𝑥 → 𝑥 ≈ 𝐴)
24		isnumi 10037	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → 𝐴 ∈ dom card)
25	22, 23, 24	syl2an 608	. . . . . 6 ⊢ ((𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥) → 𝐴 ∈ dom card)
26	25	rexlimiva 3158	. . . . 5 ⊢ (∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥 → 𝐴 ∈ dom card)
27	26	con3i 155	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥)
28		isfin7 10389	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
29	27, 28	imbitrrid 249	. . 3 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ dom card → 𝐴 ∈ FinVII))
30		fin17 10482	. . . 4 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinVII)
31	30	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin → 𝐴 ∈ FinVII))
32	29, 31	jad 189	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ dom card → 𝐴 ∈ Fin) → 𝐴 ∈ FinVII))
33	21, 32	impbid2 229	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  ∃wrex 3089   ∖ cdif 4047   class class class wbr 5105  dom cdm 5654  Oncon0 6368  ωcom 7884   ≈ cen 8981  Fincfn 8984  cardccrd 10026  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-pow 5330  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-int 4910  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-er 8728  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 10030  df-fin7 10379

This theorem is used by:  fin71num  10485  dffin7-2  10486