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Theorem scottex 9880
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 5308 . . . 4 βˆ… ∈ V
2 eleq1 2822 . . . 4 (𝐴 = βˆ… β†’ (𝐴 ∈ V ↔ βˆ… ∈ V))
31, 2mpbiri 258 . . 3 (𝐴 = βˆ… β†’ 𝐴 ∈ V)
4 rabexg 5332 . . 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 4346 . . 3 (Β¬ 𝐴 = βˆ… ↔ βˆƒπ‘¦ 𝑦 ∈ 𝐴)
7 nfra1 3282 . . . . . 6 β„²π‘¦βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)
8 nfcv 2904 . . . . . 6 Ⅎ𝑦𝐴
97, 8nfrabw 3469 . . . . 5 Ⅎ𝑦{π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)}
109nfel1 2920 . . . 4 Ⅎ𝑦{π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V
11 rsp 3245 . . . . . . . 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 482 . . . . . 6 ((𝑦 ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ (βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))
1413ss2rabdv 4074 . . . . 5 (𝑦 ∈ 𝐴 β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} βŠ† {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)})
15 rankon 9790 . . . . . . . 8 (rankβ€˜π‘¦) ∈ On
16 fveq2 6892 . . . . . . . . . . . 12 (π‘₯ = 𝑀 β†’ (rankβ€˜π‘₯) = (rankβ€˜π‘€))
1716sseq1d 4014 . . . . . . . . . . 11 (π‘₯ = 𝑀 β†’ ((rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) ↔ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)))
1817elrab 3684 . . . . . . . . . 10 (𝑀 ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ↔ (𝑀 ∈ 𝐴 ∧ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)))
1918simprbi 498 . . . . . . . . 9 (𝑀 ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} β†’ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦))
2019rgen 3064 . . . . . . . 8 βˆ€π‘€ ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)
21 sseq2 4009 . . . . . . . . . 10 (𝑧 = (rankβ€˜π‘¦) β†’ ((rankβ€˜π‘€) βŠ† 𝑧 ↔ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)))
2221ralbidv 3178 . . . . . . . . 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 3613 . . . . . . . 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 691 . . . . . . 7 βˆƒπ‘§ ∈ On βˆ€π‘€ ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} (rankβ€˜π‘€) βŠ† 𝑧
25 bndrank 9836 . . . . . . 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 5322 . . . . 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 2211 . . 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 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323  Oncon0 6365  β€˜cfv 6544  rankcrnk 9758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-reg 9587  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-r1 9759  df-rank 9760
This theorem is referenced by:  scottexs  9882  cplem2  9885  kardex  9889  scottexf  37036  scottex2  43004
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