Theorem isfin7-2 10290

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 10195	. . . 4 ⊢ (𝐴 ∈ FinVII → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
2	1	ibi 267	. . 3 ⊢ (𝐴 ∈ FinVII → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥)
3		isnum2 9841	. . . . 5 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
4		ensym 8928	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ≈ 𝑥)
5		simprl 770	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝑥 ∈ On)
6		enfi 9101	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≈ 𝑥 → (𝐴 ∈ Fin ↔ 𝑥 ∈ Fin))
7		onfin 9129	. . . . . . . . . . . . . . 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 3914	. . . . . . . . . 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 683	. . . . . . . 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 3139	. . . . . 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 4082	. . . . . . 7 ⊢ (𝑥 ∈ (On ∖ ω) → 𝑥 ∈ On)
23		ensym 8928	. . . . . . 7 ⊢ (𝐴 ≈ 𝑥 → 𝑥 ≈ 𝐴)
24		isnumi 9842	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → 𝐴 ∈ dom card)
25	22, 23, 24	syl2an 596	. . . . . 6 ⊢ ((𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥) → 𝐴 ∈ dom card)
26	25	rexlimiva 3122	. . . . 5 ⊢ (∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥 → 𝐴 ∈ dom card)
27	26	con3i 154	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥)
28		isfin7 10195	. . . 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 10288	. . . 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 2109  ∃wrex 3053   ∖ cdif 3900   class class class wbr 5092  dom cdm 5619  Oncon0 6307  ωcom 7799   ≈ cen 8869  Fincfn 8872  cardccrd 9831  Fin^VIIcfin7 10178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-om 7800  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-card 9835  df-fin7 10185

This theorem is referenced by:  fin71num  10291  dffin7-2  10292