Theorem fineqvlem 8784

Description: Lemma for fineqv 8785. (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 5252	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2	1	adantr 484	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 𝒫 𝐴 ∈ V)
3	2	pwexd 5253	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 𝒫 𝒫 𝐴 ∈ V)
4		ssrab2 3987	. . . . 5 ⊢ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ⊆ 𝒫 𝐴
5		elpw2g 5219	. . . . . 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 261	. . . 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 8783	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑏 ∈ ω ∃𝑒(𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏))
10	9	r19.21bi 3138	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑏 ∈ ω) → ∃𝑒(𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏))
11	10	ad2ant2lr 747	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ∃𝑒(𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏))
12		velpw 4503	. . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝒫 𝐴 ↔ 𝑒 ⊆ 𝐴)
13	12	biimpri 231	. . . . . . . . . 10 ⊢ (𝑒 ⊆ 𝐴 → 𝑒 ∈ 𝒫 𝐴)
14	13	anim1i 617	. . . . . . . . 9 ⊢ ((𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏) → (𝑒 ∈ 𝒫 𝐴 ∧ 𝑒 ≈ 𝑏))
15		breq1 5040	. . . . . . . . . 10 ⊢ (𝑑 = 𝑒 → (𝑑 ≈ 𝑏 ↔ 𝑒 ≈ 𝑏))
16	15	elrab 3605	. . . . . . . . 9 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ↔ (𝑒 ∈ 𝒫 𝐴 ∧ 𝑒 ≈ 𝑏))
17	14, 16	sylibr 237	. . . . . . . 8 ⊢ ((𝑒 ⊆ 𝐴 ∧ 𝑒 ≈ 𝑏) → 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏})
18	17	eximi 1837	. . . . . . 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 2841	. . . . . . . . 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 252	. . . . . . . 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 485	. . . . . . 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 500	. . . . . . . . . 10 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} → 𝑒 ≈ 𝑏)
24		breq1 5040	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑒 → (𝑑 ≈ 𝑐 ↔ 𝑒 ≈ 𝑐))
25	24	elrab 3605	. . . . . . . . . . 11 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} ↔ (𝑒 ∈ 𝒫 𝐴 ∧ 𝑒 ≈ 𝑐))
26	25	simprbi 500	. . . . . . . . . 10 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑒 ≈ 𝑐)
27		ensym 8590	. . . . . . . . . . 11 ⊢ (𝑒 ≈ 𝑏 → 𝑏 ≈ 𝑒)
28		entr 8593	. . . . . . . . . . 11 ⊢ ((𝑏 ≈ 𝑒 ∧ 𝑒 ≈ 𝑐) → 𝑏 ≈ 𝑐)
29	27, 28	sylan 583	. . . . . . . . . 10 ⊢ ((𝑒 ≈ 𝑏 ∧ 𝑒 ≈ 𝑐) → 𝑏 ≈ 𝑐)
30	23, 26, 29	syl2an 598	. . . . . . . . 9 ⊢ ((𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐}) → 𝑏 ≈ 𝑐)
31	30	ex 416	. . . . . . . 8 ⊢ (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} → (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑏 ≈ 𝑐))
32	31	adantl 485	. . . . . . 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 8736	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏 ≈ 𝑐 ↔ 𝑏 = 𝑐))
34	33	biimpd 232	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏 ≈ 𝑐 → 𝑏 = 𝑐))
35	34	ad2antlr 726	. . . . . . 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 1937	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} → 𝑏 = 𝑐))
38		breq2 5041	. . . . . 6 ⊢ (𝑏 = 𝑐 → (𝑑 ≈ 𝑏 ↔ 𝑑 ≈ 𝑐))
39	38	rabbidv 3393	. . . . 5 ⊢ (𝑏 = 𝑐 → {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐})
40	37, 39	impbid1 228	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ({𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑏} = {𝑑 ∈ 𝒫 𝐴 ∣ 𝑑 ≈ 𝑐} ↔ 𝑏 = 𝑐))
41	40	ex 416	. . 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 8582	. 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 209   ∧ wa 399   = wceq 1539  ∃wex 1782   ∈ wcel 2112  {crab 3075  Vcvv 3410   ⊆ wss 3861  𝒫 cpw 4498   class class class wbr 5037  ωcom 7586   ≈ cen 8538   ≼ cdom 8539  Fincfn 8541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-om 7587  df-er 8306  df-en 8542  df-dom 8543  df-fin 8545

This theorem is referenced by:  fineqv  8785  isfin1-2  9859