Theorem isfin7-2 10309

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 10214	. . . 4 ⊢ (𝐴 ∈ FinVII → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
2	1	ibi 268	. . 3 ⊢ (𝐴 ∈ FinVII → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥)
3		isnum2 9860	. . . . 5 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
4		ensym 8940	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ≈ 𝑥)
5		simprl 776	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝑥 ∈ On)
6		enfi 9111	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≈ 𝑥 → (𝐴 ∈ Fin ↔ 𝑥 ∈ Fin))
7		onfin 9139	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ On → (𝑥 ∈ Fin ↔ 𝑥 ∈ ω))
8	6, 7	sylan9bbr 515	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝐴 ≈ 𝑥) → (𝐴 ∈ Fin ↔ 𝑥 ∈ ω))
9	8	biimprd 249	. . . . . . . . . . . . 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 408	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → ¬ 𝑥 ∈ ω)
12	5, 11	eldifd 3894	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝑥 ∈ (On ∖ ω))
13		simprr 778	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → 𝐴 ≈ 𝑥)
14	12, 13	jca 516	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴 ≈ 𝑥)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥))
15	4, 14	sylanr2 689	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝑥 ≈ 𝐴)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥))
16	15	ex 413	. . . . . . 7 ⊢ (¬ 𝐴 ∈ Fin → ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥)))
17	16	reximdv2 3149	. . . . . 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 218	. . . 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 4061	. . . . . . 7 ⊢ (𝑥 ∈ (On ∖ ω) → 𝑥 ∈ On)
23		ensym 8940	. . . . . . 7 ⊢ (𝐴 ≈ 𝑥 → 𝑥 ≈ 𝐴)
24		isnumi 9861	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → 𝐴 ∈ dom card)
25	22, 23, 24	syl2an 602	. . . . . 6 ⊢ ((𝑥 ∈ (On ∖ ω) ∧ 𝐴 ≈ 𝑥) → 𝐴 ∈ dom card)
26	25	rexlimiva 3132	. . . . 5 ⊢ (∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥 → 𝐴 ∈ dom card)
27	26	con3i 154	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥)
28		isfin7 10214	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴 ≈ 𝑥))
29	27, 28	imbitrrid 247	. . 3 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ dom card → 𝐴 ∈ FinVII))
30		fin17 10307	. . . 4 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinVII)
31	30	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin → 𝐴 ∈ FinVII))
32	29, 31	jad 188	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ dom card → 𝐴 ∈ Fin) → 𝐴 ∈ FinVII))
33	21, 32	impbid2 227	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  ∃wrex 3063   ∖ cdif 3880   class class class wbr 5072  dom cdm 5618  Oncon0 6310  ωcom 7806   ≈ cen 8880  Fincfn 8883  cardccrd 9850  Fin^VIIcfin7 10197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-om 7807  df-1o 8395  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9854  df-fin7 10204

This theorem is referenced by:  fin71num  10310  dffin7-2  10311