Theorem fineqvlem 9164

Description: Lemma for fineqv 9165. (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 5321	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 𝒫 𝐴 ∈ V)
3	2	pwexd 5322	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 𝒫 𝒫 𝐴 ∈ V)
4		ssrab2 4030	. . . . 5 ⊢ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ⊆ 𝒫 𝐴
5		elpw2g 5276	. . . . . 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 258	. . . 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 9163	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑏 ∈ ω ∃𝑒(𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏))
10	9	r19.21bi 3226	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑏 ∈ ω) → ∃𝑒(𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏))
11	10	ad2ant2lr 748	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ∃𝑒(𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏))
12		velpw 4557	. . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝒫 𝐴 ↔ 𝑒 ⊆ 𝐴)
13	12	biimpri 228	. . . . . . . . . 10 ⊢ (𝑒 ⊆ 𝐴 → 𝑒 ∈ 𝒫 𝐴)
14	13	anim1i 615	. . . . . . . . 9 ⊢ ((𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏) → (𝑒 ∈ 𝒫 𝐴 ∧ 𝑒 ≈ 𝑏))
15		breq1 5099	. . . . . . . . . 10 ⊢ (𝑑 = 𝑒 → (𝑑 ≈ 𝑏 ↔ 𝑒 ≈ 𝑏))
16	15	elrab 3644	. . . . . . . . 9 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ↔ (𝑒 ∈ 𝒫 𝐴 ∧ 𝑒 ≈ 𝑏))
17	14, 16	sylibr 234	. . . . . . . 8 ⊢ ((𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏) → 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏})
18	17	eximi 1836	. . . . . . 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 2823	. . . . . . . . 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 249	. . . . . . . 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 481	. . . . . . 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 496	. . . . . . . . . 10 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} → 𝑒 ≈ 𝑏)
24		breq1 5099	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑒 → (𝑑 ≈ 𝑐 ↔ 𝑒 ≈ 𝑐))
25	24	elrab 3644	. . . . . . . . . . 11 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} ↔ (𝑒 ∈ 𝒫 𝐴 ∧ 𝑒 ≈ 𝑐))
26	25	simprbi 496	. . . . . . . . . 10 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑒 ≈ 𝑐)
27		ensym 8938	. . . . . . . . . . 11 ⊢ (𝑒 ≈ 𝑏 → 𝑏 ≈ 𝑒)
28		entr 8941	. . . . . . . . . . 11 ⊢ ((𝑏 ≈ 𝑒 ∧ 𝑒 ≈ 𝑐) → 𝑏 ≈ 𝑐)
29	27, 28	sylan 580	. . . . . . . . . 10 ⊢ ((𝑒 ≈ 𝑏 ∧ 𝑒 ≈ 𝑐) → 𝑏 ≈ 𝑐)
30	23, 26, 29	syl2an 596	. . . . . . . . 9 ⊢ ((𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐}) → 𝑏 ≈ 𝑐)
31	30	ex 412	. . . . . . . 8 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} → (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑏 ≈ 𝑐))
32	31	adantl 481	. . . . . . 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 9128	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏 ≈ 𝑐 ↔ 𝑏 = 𝑐))
34	33	biimpd 229	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏 ≈ 𝑐 → 𝑏 = 𝑐))
35	34	ad2antlr 727	. . . . . . 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 1936	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑏 = 𝑐))
38		breq2 5100	. . . . . 6 ⊢ (𝑏 = 𝑐 → (𝑑 ≈ 𝑏 ↔ 𝑑 ≈ 𝑐))
39	38	rabbidv 3404	. . . . 5 ⊢ (𝑏 = 𝑐 → {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐})
40	37, 39	impbid1 225	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} ↔ 𝑏 = 𝑐))
41	40	ex 412	. . 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 8928	. 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 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {crab 3397  Vcvv 3438   ⊆ wss 3899  𝒫 cpw 4552   class class class wbr 5096  ωcom 7806   ≈ cen 8878   ≼ cdom 8879  Fincfn 8881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-om 7807  df-1o 8395  df-er 8633  df-en 8882  df-dom 8883  df-fin 8885

This theorem is referenced by:  fineqv  9165  isfin1-2  10293