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Theorem scottex 9360
 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 5181 . . . 4 ∅ ∈ V
2 eleq1 2839 . . . 4 (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
31, 2mpbiri 261 . . 3 (𝐴 = ∅ → 𝐴 ∈ V)
4 rabexg 5205 . . 3 (𝐴 ∈ V → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
53, 4syl 17 . 2 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
6 neq0 4246 . . 3 𝐴 = ∅ ↔ ∃𝑦 𝑦𝐴)
7 nfra1 3147 . . . . . 6 𝑦𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)
8 nfcv 2919 . . . . . 6 𝑦𝐴
97, 8nfrabw 3303 . . . . 5 𝑦{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
109nfel1 2935 . . . 4 𝑦{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
11 rsp 3134 . . . . . . . 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 3114 . . . . . 6 (𝑦𝐴 → ∀𝑥𝐴 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
14 ss2rab 3977 . . . . . 6 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ ∀𝑥𝐴 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
1513, 14sylibr 237 . . . . 5 (𝑦𝐴 → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ⊆ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)})
16 rankon 9270 . . . . . . . 8 (rank‘𝑦) ∈ On
17 fveq2 6663 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (rank‘𝑥) = (rank‘𝑤))
1817sseq1d 3925 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
1918elrab 3604 . . . . . . . . . 10 (𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ (𝑤𝐴 ∧ (rank‘𝑤) ⊆ (rank‘𝑦)))
2019simprbi 500 . . . . . . . . 9 (𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} → (rank‘𝑤) ⊆ (rank‘𝑦))
2120rgen 3080 . . . . . . . 8 𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)
22 sseq2 3920 . . . . . . . . . 10 (𝑧 = (rank‘𝑦) → ((rank‘𝑤) ⊆ 𝑧 ↔ (rank‘𝑤) ⊆ (rank‘𝑦)))
2322ralbidv 3126 . . . . . . . . 9 (𝑧 = (rank‘𝑦) → (∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧 ↔ ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ (rank‘𝑦)))
2423rspcev 3543 . . . . . . . 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 691 . . . . . . 7 𝑧 ∈ On ∀𝑤 ∈ {𝑥𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑤) ⊆ 𝑧
26 bndrank 9316 . . . . . . 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 5195 . . . . 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 2215 . . 3 (∃𝑦 𝑦𝐴 → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
316, 30sylbi 220 . 2 𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
325, 31pm2.61i 185 1 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ⊆ wss 3860  ∅c0 4227  Oncon0 6174  ‘cfv 6340  rankcrnk 9238 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-reg 9102  ax-inf2 9150 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-om 7586  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-r1 9239  df-rank 9240 This theorem is referenced by:  scottexs  9362  cplem2  9365  kardex  9369  scottexf  35921  scottex2  41371
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