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Theorem scottex 9882
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 {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V
Distinct variable group:   π‘₯,𝑦,𝐴

Proof of Theorem scottex
Dummy variables 𝑧 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 5306 . . . 4 βˆ… ∈ V
2 eleq1 2819 . . . 4 (𝐴 = βˆ… β†’ (𝐴 ∈ V ↔ βˆ… ∈ V))
31, 2mpbiri 257 . . 3 (𝐴 = βˆ… β†’ 𝐴 ∈ V)
4 rabexg 5330 . . 3 (𝐴 ∈ V β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V)
53, 4syl 17 . 2 (𝐴 = βˆ… β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V)
6 neq0 4344 . . 3 (Β¬ 𝐴 = βˆ… ↔ βˆƒπ‘¦ 𝑦 ∈ 𝐴)
7 nfra1 3279 . . . . . 6 β„²π‘¦βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)
8 nfcv 2901 . . . . . 6 Ⅎ𝑦𝐴
97, 8nfrabw 3466 . . . . 5 Ⅎ𝑦{π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)}
109nfel1 2917 . . . 4 Ⅎ𝑦{π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V
11 rsp 3242 . . . . . . . 8 (βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) β†’ (𝑦 ∈ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))
1211com12 32 . . . . . . 7 (𝑦 ∈ 𝐴 β†’ (βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))
1312adantr 479 . . . . . 6 ((𝑦 ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ (βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))
1413ss2rabdv 4072 . . . . 5 (𝑦 ∈ 𝐴 β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} βŠ† {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)})
15 rankon 9792 . . . . . . . 8 (rankβ€˜π‘¦) ∈ On
16 fveq2 6890 . . . . . . . . . . . 12 (π‘₯ = 𝑀 β†’ (rankβ€˜π‘₯) = (rankβ€˜π‘€))
1716sseq1d 4012 . . . . . . . . . . 11 (π‘₯ = 𝑀 β†’ ((rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) ↔ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)))
1817elrab 3682 . . . . . . . . . 10 (𝑀 ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ↔ (𝑀 ∈ 𝐴 ∧ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)))
1918simprbi 495 . . . . . . . . 9 (𝑀 ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} β†’ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦))
2019rgen 3061 . . . . . . . 8 βˆ€π‘€ ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)
21 sseq2 4007 . . . . . . . . . 10 (𝑧 = (rankβ€˜π‘¦) β†’ ((rankβ€˜π‘€) βŠ† 𝑧 ↔ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)))
2221ralbidv 3175 . . . . . . . . 9 (𝑧 = (rankβ€˜π‘¦) β†’ (βˆ€π‘€ ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} (rankβ€˜π‘€) βŠ† 𝑧 ↔ βˆ€π‘€ ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)))
2322rspcev 3611 . . . . . . . 8 (((rankβ€˜π‘¦) ∈ On ∧ βˆ€π‘€ ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)) β†’ βˆƒπ‘§ ∈ On βˆ€π‘€ ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} (rankβ€˜π‘€) βŠ† 𝑧)
2415, 20, 23mp2an 688 . . . . . . 7 βˆƒπ‘§ ∈ On βˆ€π‘€ ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} (rankβ€˜π‘€) βŠ† 𝑧
25 bndrank 9838 . . . . . . 7 (βˆƒπ‘§ ∈ On βˆ€π‘€ ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} (rankβ€˜π‘€) βŠ† 𝑧 β†’ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V)
2624, 25ax-mp 5 . . . . . 6 {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V
2726ssex 5320 . . . . 5 ({π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} βŠ† {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V)
2814, 27syl 17 . . . 4 (𝑦 ∈ 𝐴 β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V)
2910, 28exlimi 2208 . . 3 (βˆƒπ‘¦ 𝑦 ∈ 𝐴 β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V)
306, 29sylbi 216 . 2 (Β¬ 𝐴 = βˆ… β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V)
315, 30pm2.61i 182 1 {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430  Vcvv 3472   βŠ† wss 3947  βˆ…c0 4321  Oncon0 6363  β€˜cfv 6542  rankcrnk 9760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-reg 9589  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-r1 9761  df-rank 9762
This theorem is referenced by:  scottexs  9884  cplem2  9887  kardex  9891  scottexf  37339  scottex2  43306
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