Theorem fineqvlem 9293

Description: Lemma for fineqv 9294. (Contributed by Mario Carneiro, 20-Jan-2013.) (Proof shortened by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Assertion

Ref	Expression
fineqvlem	⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴)

Proof of Theorem fineqvlem

Dummy variables 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 5382	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2	1	adantr 479	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 𝒫 𝐴 ∈ V)
3	2	pwexd 5383	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 𝒫 𝒫 𝐴 ∈ V)
4		ssrab2 4077	. . . . 5 ⊢ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ⊆ 𝒫 𝐴
5		elpw2g 5350	. . . . . 6 ⊢ (𝒫 𝐴 ∈ V → ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ∈ 𝒫 𝒫 𝐴 ↔ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ⊆ 𝒫 𝐴))
6	2, 5	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ∈ 𝒫 𝒫 𝐴 ↔ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ⊆ 𝒫 𝐴))
7	4, 6	mpbiri 257	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ∈ 𝒫 𝒫 𝐴)
8	7	a1d 25	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (𝑏 ∈ ω → {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ∈ 𝒫 𝒫 𝐴))
9		isinf 9291	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑏 ∈ ω ∃𝑒(𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏))
10	9	r19.21bi 3246	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑏 ∈ ω) → ∃𝑒(𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏))
11	10	ad2ant2lr 746	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ∃𝑒(𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏))
12		velpw 4611	. . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝒫 𝐴 ↔ 𝑒 ⊆ 𝐴)
13	12	biimpri 227	. . . . . . . . . 10 ⊢ (𝑒 ⊆ 𝐴 → 𝑒 ∈ 𝒫 𝐴)
14	13	anim1i 613	. . . . . . . . 9 ⊢ ((𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏) → (𝑒 ∈ 𝒫 𝐴 ∧ 𝑒 ≈ 𝑏))
15		breq1 5155	. . . . . . . . . 10 ⊢ (𝑑 = 𝑒 → (𝑑 ≈ 𝑏 ↔ 𝑒 ≈ 𝑏))
16	15	elrab 3684	. . . . . . . . 9 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ↔ (𝑒 ∈ 𝒫 𝐴 ∧ 𝑒 ≈ 𝑏))
17	14, 16	sylibr 233	. . . . . . . 8 ⊢ ((𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏) → 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏})
18	17	eximi 1829	. . . . . . 7 ⊢ (∃𝑒(𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏) → ∃𝑒 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏})
19	11, 18	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ∃𝑒 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏})
20		eleq2 2818	. . . . . . . . 9 ⊢ ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ↔ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐}))
21	20	biimpcd 248	. . . . . . . 8 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} → ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐}))
22	21	adantl 480	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏}) → ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐}))
23	16	simprbi 495	. . . . . . . . . 10 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} → 𝑒 ≈ 𝑏)
24		breq1 5155	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑒 → (𝑑 ≈ 𝑐 ↔ 𝑒 ≈ 𝑐))
25	24	elrab 3684	. . . . . . . . . . 11 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} ↔ (𝑒 ∈ 𝒫 𝐴 ∧ 𝑒 ≈ 𝑐))
26	25	simprbi 495	. . . . . . . . . 10 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑒 ≈ 𝑐)
27		ensym 9030	. . . . . . . . . . 11 ⊢ (𝑒 ≈ 𝑏 → 𝑏 ≈ 𝑒)
28		entr 9033	. . . . . . . . . . 11 ⊢ ((𝑏 ≈ 𝑒 ∧ 𝑒 ≈ 𝑐) → 𝑏 ≈ 𝑐)
29	27, 28	sylan 578	. . . . . . . . . 10 ⊢ ((𝑒 ≈ 𝑏 ∧ 𝑒 ≈ 𝑐) → 𝑏 ≈ 𝑐)
30	23, 26, 29	syl2an 594	. . . . . . . . 9 ⊢ ((𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐}) → 𝑏 ≈ 𝑐)
31	30	ex 411	. . . . . . . 8 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} → (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑏 ≈ 𝑐))
32	31	adantl 480	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏}) → (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑏 ≈ 𝑐))
33		nneneq 9240	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏 ≈ 𝑐 ↔ 𝑏 = 𝑐))
34	33	biimpd 228	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏 ≈ 𝑐 → 𝑏 = 𝑐))
35	34	ad2antlr 725	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏}) → (𝑏 ≈ 𝑐 → 𝑏 = 𝑐))
36	22, 32, 35	3syld 60	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏}) → ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑏 = 𝑐))
37	19, 36	exlimddv 1930	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑏 = 𝑐))
38		breq2 5156	. . . . . 6 ⊢ (𝑏 = 𝑐 → (𝑑 ≈ 𝑏 ↔ 𝑑 ≈ 𝑐))
39	38	rabbidv 3438	. . . . 5 ⊢ (𝑏 = 𝑐 → {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐})
40	37, 39	impbid1 224	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} ↔ 𝑏 = 𝑐))
41	40	ex 411	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} ↔ 𝑏 = 𝑐)))
42	8, 41	dom2d 9020	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (𝒫 𝒫 𝐴 ∈ V → ω ≼ 𝒫 𝒫 𝐴))
43	3, 42	mpd 15	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {crab 3430  Vcvv 3473   ⊆ wss 3949  𝒫 cpw 4606   class class class wbr 5152  ωcom 7876   ≈ cen 8967   ≼ cdom 8968  Fincfn 8970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7877  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-fin 8974

This theorem is referenced by:  fineqv  9294  isfin1-2  10416