Theorem isfin7-2 9975

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 9880	. . . 4 ⊢ (𝐴 ∈ FinVII → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
2	1	ibi 270	. . 3 ⊢ (𝐴 ∈ FinVII → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥)
3		isnum2 9526	. . . . 5 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
4		ensym 8655	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ≈ 𝑥)
5		simprl 771	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝑥 ∈ On)
6		enfi 8842	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≈ 𝑥 → (𝐴 ∈ Fin ↔ 𝑥 ∈ Fin))
7		onfin 8846	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ On → (𝑥 ∈ Fin ↔ 𝑥 ∈ ω))
8	6, 7	sylan9bbr 514	. . . . . . . . . . . . . 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 155	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → (¬ 𝐴 ∈ Fin → ¬ 𝑥 ∈ ω))
11	10	impcom 411	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → ¬ 𝑥 ∈ ω)
12	5, 11	eldifd 3864	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝑥 ∈ (On ∖ ω))
13		simprr 773	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝐴 ≈ 𝑥)
14	12, 13	jca 515	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥))
15	4, 14	sylanr2 683	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝑥 ≈ 𝐴)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥))
16	15	ex 416	. . . . . . 7 ⊢ (¬ 𝐴 ∈ Fin → ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥)))
17	16	reximdv2 3180	. . . . . 6 ⊢ (¬ 𝐴 ∈ Fin → (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
18	17	com12 32	. . . . 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 147	. . 3 ⊢ (𝐴 ∈ dom card → (¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥 → 𝐴 ∈ Fin))
21	2, 20	syl5com 31	. 2 ⊢ (𝐴 ∈ FinVII → (𝐴 ∈ dom card → 𝐴 ∈ Fin))
22		eldifi 4027	. . . . . . 7 ⊢ (𝑥 ∈ (On ∖ ω) → 𝑥 ∈ On)
23		ensym 8655	. . . . . . 7 ⊢ (𝐴 ≈ 𝑥 → 𝑥 ≈ 𝐴)
24		isnumi 9527	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → 𝐴 ∈ dom card)
25	22, 23, 24	syl2an 599	. . . . . 6 ⊢ ((𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥) → 𝐴 ∈ dom card)
26	25	rexlimiva 3190	. . . . 5 ⊢ (∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥 → 𝐴 ∈ dom card)
27	26	con3i 157	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥)
28		isfin7 9880	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
29	27, 28	syl5ibr 249	. . 3 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ dom card → 𝐴 ∈ FinVII))
30		fin17 9973	. . . 4 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinVII)
31	30	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin → 𝐴 ∈ FinVII))
32	29, 31	jad 190	. 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)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2112  ∃wrex 3052   ∖ cdif 3850   class class class wbr 5039  dom cdm 5536  Oncon0 6191  ωcom 7622   ≈ cen 8601  Fincfn 8604  cardccrd 9516  Fin^VIIcfin7 9863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-om 7623  df-1o 8180  df-er 8369  df-en 8605  df-dom 8606  df-sdom 8607  df-fin 8608  df-card 9520  df-fin7 9870

This theorem is referenced by:  fin71num  9976  dffin7-2  9977