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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 {𝑥𝐴 ∣ ∀𝑦𝐴 (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.
StepHypRef Expression
1 0ex 5068 . . . 4 ∅ ∈ V
2 eleq1 2853 . . . 4 (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
31, 2mpbiri 250 . . 3 (𝐴 = ∅ → 𝐴 ∈ V)
4 rabexg 5090 . . 3 (𝐴 ∈ V → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
53, 4syl 17 . 2 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
6 neq0 4195 . . 3 𝐴 = ∅ ↔ ∃𝑦 𝑦𝐴)
7 nfra1 3169 . . . . . 6 𝑦𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)
8 nfcv 2932 . . . . . 6 𝑦𝐴
97, 8nfrab 3325 . . . . 5 𝑦{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
109nfel1 2946 . . . 4 𝑦{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
11 rsp 3155 . . . . . . . 8 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
1211com12 32 . . . . . . 7 (𝑦𝐴 → (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
1312ralrimivw 3133 . . . . . 6 (𝑦𝐴 → ∀𝑥𝐴 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
14 ss2rab 3937 . . . . . 6 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ ∀𝑥𝐴 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
1513, 14sylibr 226 . . . . 5 (𝑦𝐴 → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)})
16 rankon 9018 . . . . . . . 8 (rank‘𝑦) ∈ On
17 fveq2 6499 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (rank‘𝑥) = (rank‘𝑤))
1817sseq1d 3888 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
1918elrab 3595 . . . . . . . . . 10 (𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ (𝑤𝐴 ∧ (rank‘𝑤) ⊆ (rank‘𝑦)))
2019simprbi 489 . . . . . . . . 9 (𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} → (rank‘𝑤) ⊆ (rank‘𝑦))
2120rgen 3098 . . . . . . . 8 𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)
22 sseq2 3883 . . . . . . . . . 10 (𝑧 = (rank‘𝑦) → ((rank‘𝑤) ⊆ 𝑧 ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
2322ralbidv 3147 . . . . . . . . 9 (𝑧 = (rank‘𝑦) → (∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧 ↔ ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)))
2423rspcev 3535 . . . . . . . 8 (((rank‘𝑦) ∈ On ∧ ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)) → ∃𝑧 ∈ On ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧)
2516, 21, 24mp2an 679 . . . . . . 7 𝑧 ∈ On ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧
26 bndrank 9064 . . . . . . 7 (∃𝑧 ∈ On ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧 → {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
2725, 26ax-mp 5 . . . . . 6 {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
2827ssex 5081 . . . . 5 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
2915, 28syl 17 . . . 4 (𝑦𝐴 → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
3010, 29exlimi 2147 . . 3 (∃𝑦 𝑦𝐴 → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
316, 30sylbi 209 . 2 𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
325, 31pm2.61i 177 1 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
Colors of variables: wff setvar class
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
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