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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 {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 5301 . . . 4 ∅ ∈ V
2 eleq1 2817 . . . 4 (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
31, 2mpbiri 258 . . 3 (𝐴 = ∅ → 𝐴 ∈ V)
4 rabexg 5327 . . 3 (𝐴 ∈ V → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
53, 4syl 17 . 2 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
6 neq0 4341 . . 3 𝐴 = ∅ ↔ ∃𝑦 𝑦𝐴)
7 nfra1 3277 . . . . . 6 𝑦𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)
8 nfcv 2899 . . . . . 6 𝑦𝐴
97, 8nfrabw 3464 . . . . 5 𝑦{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
109nfel1 2915 . . . 4 𝑦{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
11 rsp 3240 . . . . . . . 8 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
1211com12 32 . . . . . . 7 (𝑦𝐴 → (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
1312adantr 480 . . . . . 6 ((𝑦𝐴𝑥𝐴) → (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
1413ss2rabdv 4069 . . . . 5 (𝑦𝐴 → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)})
15 rankon 9812 . . . . . . . 8 (rank‘𝑦) ∈ On
16 fveq2 6891 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (rank‘𝑥) = (rank‘𝑤))
1716sseq1d 4009 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
1817elrab 3681 . . . . . . . . . 10 (𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ (𝑤𝐴 ∧ (rank‘𝑤) ⊆ (rank‘𝑦)))
1918simprbi 496 . . . . . . . . 9 (𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} → (rank‘𝑤) ⊆ (rank‘𝑦))
2019rgen 3059 . . . . . . . 8 𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)
21 sseq2 4004 . . . . . . . . . 10 (𝑧 = (rank‘𝑦) → ((rank‘𝑤) ⊆ 𝑧 ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
2221ralbidv 3173 . . . . . . . . 9 (𝑧 = (rank‘𝑦) → (∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧 ↔ ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)))
2322rspcev 3608 . . . . . . . 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)
2415, 20, 23mp2an 691 . . . . . . 7 𝑧 ∈ On ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧
25 bndrank 9858 . . . . . . 7 (∃𝑧 ∈ On ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧 → {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
2624, 25ax-mp 5 . . . . . 6 {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
2726ssex 5315 . . . . 5 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
2814, 27syl 17 . . . 4 (𝑦𝐴 → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
2910, 28exlimi 2206 . . 3 (∃𝑦 𝑦𝐴 → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
306, 29sylbi 216 . 2 𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
315, 30pm2.61i 182 1 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
Colors of variables: wff setvar class
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
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