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

Theorem scottex 9829
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 5268 . . . 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 5292 . . 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 4309 . . 3 (Β¬ 𝐴 = βˆ… ↔ βˆƒπ‘¦ 𝑦 ∈ 𝐴)
7 nfra1 3266 . . . . . 6 β„²π‘¦βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)
8 nfcv 2904 . . . . . 6 Ⅎ𝑦𝐴
97, 8nfrabw 3442 . . . . 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 3229 . . . . . . . 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 4037 . . . . 5 (𝑦 ∈ 𝐴 β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} βŠ† {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)})
15 rankon 9739 . . . . . . . 8 (rankβ€˜π‘¦) ∈ On
16 fveq2 6846 . . . . . . . . . . . 12 (π‘₯ = 𝑀 β†’ (rankβ€˜π‘₯) = (rankβ€˜π‘€))
1716sseq1d 3979 . . . . . . . . . . 11 (π‘₯ = 𝑀 β†’ ((rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) ↔ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)))
1817elrab 3649 . . . . . . . . . 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 3063 . . . . . . . 8 βˆ€π‘€ ∈ {π‘₯ ∈ 𝐴 ∣ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)
21 sseq2 3974 . . . . . . . . . 10 (𝑧 = (rankβ€˜π‘¦) β†’ ((rankβ€˜π‘€) βŠ† 𝑧 ↔ (rankβ€˜π‘€) βŠ† (rankβ€˜π‘¦)))
2221ralbidv 3171 . . . . . . . . 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 3583 . . . . . . . 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 9785 . . . . . . 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 5282 . . . . 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 3061  βˆƒwrex 3070  {crab 3406  Vcvv 3447   βŠ† wss 3914  βˆ…c0 4286  Oncon0 6321  β€˜cfv 6500  rankcrnk 9707
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-reg 9536  ax-inf2 9585
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-r1 9708  df-rank 9709
This theorem is referenced by:  scottexs  9831  cplem2  9834  kardex  9838  scottexf  36677  scottex2  42617
  Copyright terms: Public domain W3C validator