Theorem scottex 9902

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 5301	. . . 4 ⊢ ∅ ∈ V
2		eleq1 2817	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
3	1, 2	mpbiri 258	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
4		rabexg 5327	. . 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 4341	. . 3 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
7		nfra1 3277	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)
8		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑦𝐴
9	7, 8	nfrabw 3464	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
10	9	nfel1 2915	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
11		rsp 3240	. . . . . . . 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	adantr 480	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
14	13	ss2rabdv 4069	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)})
15		rankon 9812	. . . . . . . 8 ⊢ (rank‘𝑦) ∈ On
16		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (rank‘𝑥) = (rank‘𝑤))
17	16	sseq1d 4009	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
18	17	elrab 3681	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ (𝑤 ∈ 𝐴 ∧ (rank‘𝑤) ⊆ (rank‘𝑦)))
19	18	simprbi 496	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} → (rank‘𝑤) ⊆ (rank‘𝑦))
20	19	rgen 3059	. . . . . . . 8 ⊢ ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)
21		sseq2 4004	. . . . . . . . . 10 ⊢ (𝑧 = (rank‘𝑦) → ((rank‘𝑤) ⊆ 𝑧 ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
22	21	ralbidv 3173	. . . . . . . . 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)))
23	22	rspcev 3608	. . . . . . . 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)
24	15, 20, 23	mp2an 691	. . . . . . 7 ⊢ ∃𝑧 ∈ On ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧
25		bndrank 9858	. . . . . . 7 ⊢ (∃𝑧 ∈ On ∀𝑤 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧 → {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
26	24, 25	ax-mp 5	. . . . . 6 ⊢ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
27	26	ssex 5315	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
28	14, 27	syl 17	. . . 4 ⊢ (𝑦 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
29	10, 28	exlimi 2206	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
30	6, 29	sylbi 216	. 2 ⊢ (¬ 𝐴 = ∅ → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
31	5, 30	pm2.61i 182	1 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∀wral 3057  ∃wrex 3066  {crab 3428  Vcvv 3470   ⊆ wss 3945  ∅c0 4318  Oncon0 6363  ‘cfv 6542  rankcrnk 9780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-reg 9609  ax-inf2 9658

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-r1 9781  df-rank 9782

This theorem is referenced by:  scottexs  9904  cplem2  9907  kardex  9911  scottexf  37635  scottex2  43676