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Theorem scottex 8786
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 4823 . . . 4 ∅ ∈ V
2 eleq1 2718 . . . 4 (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
31, 2mpbiri 248 . . 3 (𝐴 = ∅ → 𝐴 ∈ V)
4 rabexg 4844 . . 3 (𝐴 ∈ V → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
53, 4syl 17 . 2 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
6 neq0 3963 . . 3 𝐴 = ∅ ↔ ∃𝑦 𝑦𝐴)
7 nfra1 2970 . . . . . 6 𝑦𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)
8 nfcv 2793 . . . . . 6 𝑦𝐴
97, 8nfrab 3153 . . . . 5 𝑦{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
109nfel1 2808 . . . 4 𝑦{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
11 rsp 2958 . . . . . . . 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 2996 . . . . . 6 (𝑦𝐴 → ∀𝑥𝐴 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
14 ss2rab 3711 . . . . . 6 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ ∀𝑥𝐴 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
1513, 14sylibr 224 . . . . 5 (𝑦𝐴 → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)})
16 rankon 8696 . . . . . . . 8 (rank‘𝑦) ∈ On
17 fveq2 6229 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (rank‘𝑥) = (rank‘𝑤))
1817sseq1d 3665 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
1918elrab 3396 . . . . . . . . . 10 (𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ (𝑤𝐴 ∧ (rank‘𝑤) ⊆ (rank‘𝑦)))
2019simprbi 479 . . . . . . . . 9 (𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} → (rank‘𝑤) ⊆ (rank‘𝑦))
2120rgen 2951 . . . . . . . 8 𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)
22 sseq2 3660 . . . . . . . . . 10 (𝑧 = (rank‘𝑦) → ((rank‘𝑤) ⊆ 𝑧 ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
2322ralbidv 3015 . . . . . . . . 9 (𝑧 = (rank‘𝑦) → (∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧 ↔ ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)))
2423rspcev 3340 . . . . . . . 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 708 . . . . . . 7 𝑧 ∈ On ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧
26 bndrank 8742 . . . . . . 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 4835 . . . . 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 2124 . . 3 (∃𝑦 𝑦𝐴 → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
316, 30sylbi 207 . 2 𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
325, 31pm2.61i 176 1 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  c0 3948  Oncon0 5761  cfv 5926  rankcrnk 8664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666
This theorem is referenced by:  scottexs  8788  cplem2  8791  kardex  8795  scottexf  34106
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