MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  scottex Structured version   Visualization version   GIF version

Theorem scottex 8745
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 4788 . . . 4 ∅ ∈ V
2 eleq1 2688 . . . 4 (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
31, 2mpbiri 248 . . 3 (𝐴 = ∅ → 𝐴 ∈ V)
4 rabexg 4810 . . 3 (𝐴 ∈ V → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
53, 4syl 17 . 2 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
6 neq0 3928 . . 3 𝐴 = ∅ ↔ ∃𝑦 𝑦𝐴)
7 nfra1 2940 . . . . . 6 𝑦𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)
8 nfcv 2763 . . . . . 6 𝑦𝐴
97, 8nfrab 3121 . . . . 5 𝑦{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
109nfel1 2778 . . . 4 𝑦{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
11 rsp 2928 . . . . . . . 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 2966 . . . . . 6 (𝑦𝐴 → ∀𝑥𝐴 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
14 ss2rab 3676 . . . . . 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 8655 . . . . . . . 8 (rank‘𝑦) ∈ On
17 fveq2 6189 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (rank‘𝑥) = (rank‘𝑤))
1817sseq1d 3630 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
1918elrab 3361 . . . . . . . . . 10 (𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ (𝑤𝐴 ∧ (rank‘𝑤) ⊆ (rank‘𝑦)))
2019simprbi 480 . . . . . . . . 9 (𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} → (rank‘𝑤) ⊆ (rank‘𝑦))
2120rgen 2921 . . . . . . . 8 𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)
22 sseq2 3625 . . . . . . . . . 10 (𝑧 = (rank‘𝑦) → ((rank‘𝑤) ⊆ 𝑧 ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
2322ralbidv 2985 . . . . . . . . 9 (𝑧 = (rank‘𝑦) → (∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧 ↔ ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)))
2423rspcev 3307 . . . . . . . 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 8701 . . . . . . 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 4800 . . . . 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 2085 . . 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 1482  wex 1703  wcel 1989  wral 2911  wrex 2912  {crab 2915  Vcvv 3198  wss 3572  c0 3913  Oncon0 5721  cfv 5886  rankcrnk 8623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-reg 8494  ax-inf2 8535
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-r1 8624  df-rank 8625
This theorem is referenced by:  scottexs  8747  cplem2  8750  kardex  8754  scottexf  33956
  Copyright terms: Public domain W3C validator