Theorem scottex 9108

Description: Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set. (Contributed by NM, 13-Oct-2003.)

Assertion

Ref	Expression
scottex	⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem scottex

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5068	. . . 4 ⊢ ∅ ∈ V
2		eleq1 2853	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
3	1, 2	mpbiri 250	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
4		rabexg 5090	. . 3 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
5	3, 4	syl 17	. 2 ⊢ (𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
6		neq0 4195	. . 3 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
7		nfra1 3169	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)
8		nfcv 2932	. . . . . 6 ⊢ Ⅎ𝑦𝐴
9	7, 8	nfrab 3325	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
10	9	nfel1 2946	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
11		rsp 3155	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (𝑦 ∈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
12	11	com12 32	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
13	12	ralrimivw 3133	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
14		ss2rab 3937	. . . . . 6 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
15	13, 14	sylibr 226	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)})
16		rankon 9018	. . . . . . . 8 ⊢ (rank‘𝑦) ∈ On
17		fveq2 6499	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (rank‘𝑥) = (rank‘𝑤))
18	17	sseq1d 3888	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
19	18	elrab 3595	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ (𝑤 ∈ 𝐴 ∧ (rank‘𝑤) ⊆ (rank‘𝑦)))
20	19	simprbi 489	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} → (rank‘𝑤) ⊆ (rank‘𝑦))
21	20	rgen 3098	. . . . . . . 8 ⊢ ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)
22		sseq2 3883	. . . . . . . . . 10 ⊢ (𝑧 = (rank‘𝑦) → ((rank‘𝑤) ⊆ 𝑧 ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
23	22	ralbidv 3147	. . . . . . . . 9 ⊢ (𝑧 = (rank‘𝑦) → (∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧 ↔ ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)))
24	23	rspcev 3535	. . . . . . . 8 ⊢ (((rank‘𝑦) ∈ On ∧ ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)) → ∃𝑧 ∈ On ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧)
25	16, 21, 24	mp2an 679	. . . . . . 7 ⊢ ∃𝑧 ∈ On ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧
26		bndrank 9064	. . . . . . 7 ⊢ (∃𝑧 ∈ On ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧 → {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
27	25, 26	ax-mp 5	. . . . . 6 ⊢ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
28	27	ssex 5081	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
29	15, 28	syl 17	. . . 4 ⊢ (𝑦 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
30	10, 29	exlimi 2147	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
31	6, 30	sylbi 209	. 2 ⊢ (¬ 𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
32	5, 31	pm2.61i 177	1 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1507  ∃wex 1742   ∈ wcel 2050  ∀wral 3088  ∃wrex 3089  {crab 3092  Vcvv 3415   ⊆ wss 3829  ∅c0 4178  Oncon0 6029  ‘cfv 6188  rankcrnk 8986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-reg 8851  ax-inf2 8898

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-om 7397  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-r1 8987  df-rank 8988

This theorem is referenced by:  scottexs  9110  cplem2  9113  kardex  9117  scottexf  34896  scottex2  39962