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Theorem scottex 9879
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 5307 . . . 4 βˆ… ∈ V
2 eleq1 2821 . . . 4 (𝐴 = βˆ… β†’ (𝐴 ∈ V ↔ βˆ… ∈ V))
31, 2mpbiri 257 . . 3 (𝐴 = βˆ… β†’ 𝐴 ∈ V)
4 rabexg 5331 . . 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 4345 . . 3 (Β¬ 𝐴 = βˆ… ↔ βˆƒπ‘¦ 𝑦 ∈ 𝐴)
7 nfra1 3281 . . . . . 6 β„²π‘¦βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)
8 nfcv 2903 . . . . . 6 Ⅎ𝑦𝐴
97, 8nfrabw 3468 . . . . 5 Ⅎ𝑦{π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)}
109nfel1 2919 . . . 4 Ⅎ𝑦{π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V
11 rsp 3244 . . . . . . . 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 481 . . . . . 6 ((𝑦 ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ (βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))
1413ss2rabdv 4073 . . . . 5 (𝑦 ∈ 𝐴 β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} βŠ† {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)})
15 rankon 9789 . . . . . . . 8 (rankβ€˜π‘¦) ∈ On
16 fveq2 6891 . . . . . . . . . . . 12 (π‘₯ = 𝑀 β†’ (rankβ€˜π‘₯) = (rankβ€˜π‘€))
1716sseq1d 4013 . . . . . . . . . . 11 (π‘₯ = 𝑀 β†’ ((rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) ↔ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)))
1817elrab 3683 . . . . . . . . . 10 (𝑀 ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ↔ (𝑀 ∈ 𝐴 ∧ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)))
1918simprbi 497 . . . . . . . . 9 (𝑀 ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} β†’ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦))
2019rgen 3063 . . . . . . . 8 βˆ€π‘€ ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)
21 sseq2 4008 . . . . . . . . . 10 (𝑧 = (rankβ€˜π‘¦) β†’ ((rankβ€˜π‘€) βŠ† 𝑧 ↔ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)))
2221ralbidv 3177 . . . . . . . . 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 3612 . . . . . . . 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 690 . . . . . . 7 βˆƒπ‘§ ∈ On βˆ€π‘€ ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} (rankβ€˜π‘€) βŠ† 𝑧
25 bndrank 9835 . . . . . . 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 5321 . . . . 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 2210 . . 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 1541  βˆƒwex 1781   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βŠ† wss 3948  βˆ…c0 4322  Oncon0 6364  β€˜cfv 6543  rankcrnk 9757
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-reg 9586  ax-inf2 9635
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-r1 9758  df-rank 9759
This theorem is referenced by:  scottexs  9881  cplem2  9884  kardex  9888  scottexf  37031  scottex2  42994
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