Theorem isfin7-2 10465

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 10370	. . . 4 ⊢ (𝐴 ∈ FinVII → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
2	1	ibi 267	. . 3 ⊢ (𝐴 ∈ FinVII → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥)
3		isnum2 10014	. . . . 5 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
4		ensym 9063	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ≈ 𝑥)
5		simprl 770	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝑥 ∈ On)
6		enfi 9253	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≈ 𝑥 → (𝐴 ∈ Fin ↔ 𝑥 ∈ Fin))
7		onfin 9293	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ On → (𝑥 ∈ Fin ↔ 𝑥 ∈ ω))
8	6, 7	sylan9bbr 510	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → (𝐴 ∈ Fin ↔ 𝑥 ∈ ω))
9	8	biimprd 248	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → (𝑥 ∈ ω → 𝐴 ∈ Fin))
10	9	con3d 152	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → (¬ 𝐴 ∈ Fin → ¬ 𝑥 ∈ ω))
11	10	impcom 407	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → ¬ 𝑥 ∈ ω)
12	5, 11	eldifd 3987	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝑥 ∈ (On ∖ ω))
13		simprr 772	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝐴 ≈ 𝑥)
14	12, 13	jca 511	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥))
15	4, 14	sylanr2 682	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝑥 ≈ 𝐴)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥))
16	15	ex 412	. . . . . . 7 ⊢ (¬ 𝐴 ∈ Fin → ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥)))
17	16	reximdv2 3170	. . . . . 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 217	. . . 4 ⊢ (𝐴 ∈ dom card → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
20	19	con1d 145	. . 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 4154	. . . . . . 7 ⊢ (𝑥 ∈ (On ∖ ω) → 𝑥 ∈ On)
23		ensym 9063	. . . . . . 7 ⊢ (𝐴 ≈ 𝑥 → 𝑥 ≈ 𝐴)
24		isnumi 10015	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → 𝐴 ∈ dom card)
25	22, 23, 24	syl2an 595	. . . . . 6 ⊢ ((𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥) → 𝐴 ∈ dom card)
26	25	rexlimiva 3153	. . . . 5 ⊢ (∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥 → 𝐴 ∈ dom card)
27	26	con3i 154	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥)
28		isfin7 10370	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
29	27, 28	imbitrrid 246	. . 3 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ dom card → 𝐴 ∈ FinVII))
30		fin17 10463	. . . 4 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinVII)
31	30	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin → 𝐴 ∈ FinVII))
32	29, 31	jad 187	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ dom card → 𝐴 ∈ Fin) → 𝐴 ∈ FinVII))
33	21, 32	impbid2 226	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∃wrex 3076   ∖ cdif 3973   class class class wbr 5166  dom cdm 5700  Oncon0 6395  ωcom 7903   ≈ cen 9000  Fincfn 9003  cardccrd 10004  Fin^VIIcfin7 10353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-fin7 10360

This theorem is referenced by:  fin71num  10466  dffin7-2  10467