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Theorem scott0 9909
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, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. 𝐴 is empty). (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
scott0 (𝐴 = βˆ… ↔ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} = βˆ…)
Distinct variable group:   π‘₯,𝑦,𝐴

Proof of Theorem scott0
StepHypRef Expression
1 rabeq 3434 . . 3 (𝐴 = βˆ… β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} = {π‘₯ ∈ βˆ… ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)})
2 rab0 4383 . . 3 {π‘₯ ∈ βˆ… ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} = βˆ…
31, 2eqtrdi 2781 . 2 (𝐴 = βˆ… β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} = βˆ…)
4 n0 4347 . . . . . . . 8 (𝐴 β‰  βˆ… ↔ βˆƒπ‘₯ π‘₯ ∈ 𝐴)
5 nfre1 3273 . . . . . . . . 9 β„²π‘₯βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯)
6 eqid 2725 . . . . . . . . . 10 (rankβ€˜π‘₯) = (rankβ€˜π‘₯)
7 rspe 3237 . . . . . . . . . 10 ((π‘₯ ∈ 𝐴 ∧ (rankβ€˜π‘₯) = (rankβ€˜π‘₯)) β†’ βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯))
86, 7mpan2 689 . . . . . . . . 9 (π‘₯ ∈ 𝐴 β†’ βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯))
95, 8exlimi 2205 . . . . . . . 8 (βˆƒπ‘₯ π‘₯ ∈ 𝐴 β†’ βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯))
104, 9sylbi 216 . . . . . . 7 (𝐴 β‰  βˆ… β†’ βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯))
11 fvex 6907 . . . . . . . . . . 11 (rankβ€˜π‘₯) ∈ V
12 eqeq1 2729 . . . . . . . . . . . 12 (𝑦 = (rankβ€˜π‘₯) β†’ (𝑦 = (rankβ€˜π‘₯) ↔ (rankβ€˜π‘₯) = (rankβ€˜π‘₯)))
1312anbi2d 628 . . . . . . . . . . 11 (𝑦 = (rankβ€˜π‘₯) β†’ ((π‘₯ ∈ 𝐴 ∧ 𝑦 = (rankβ€˜π‘₯)) ↔ (π‘₯ ∈ 𝐴 ∧ (rankβ€˜π‘₯) = (rankβ€˜π‘₯))))
1411, 13spcev 3591 . . . . . . . . . 10 ((π‘₯ ∈ 𝐴 ∧ (rankβ€˜π‘₯) = (rankβ€˜π‘₯)) β†’ βˆƒπ‘¦(π‘₯ ∈ 𝐴 ∧ 𝑦 = (rankβ€˜π‘₯)))
1514eximi 1829 . . . . . . . . 9 (βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ (rankβ€˜π‘₯) = (rankβ€˜π‘₯)) β†’ βˆƒπ‘₯βˆƒπ‘¦(π‘₯ ∈ 𝐴 ∧ 𝑦 = (rankβ€˜π‘₯)))
16 excom 2151 . . . . . . . . 9 (βˆƒπ‘¦βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ 𝑦 = (rankβ€˜π‘₯)) ↔ βˆƒπ‘₯βˆƒπ‘¦(π‘₯ ∈ 𝐴 ∧ 𝑦 = (rankβ€˜π‘₯)))
1715, 16sylibr 233 . . . . . . . 8 (βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ (rankβ€˜π‘₯) = (rankβ€˜π‘₯)) β†’ βˆƒπ‘¦βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ 𝑦 = (rankβ€˜π‘₯)))
18 df-rex 3061 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯) ↔ βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ (rankβ€˜π‘₯) = (rankβ€˜π‘₯)))
19 df-rex 3061 . . . . . . . . 9 (βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯) ↔ βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ 𝑦 = (rankβ€˜π‘₯)))
2019exbii 1842 . . . . . . . 8 (βˆƒπ‘¦βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯) ↔ βˆƒπ‘¦βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ 𝑦 = (rankβ€˜π‘₯)))
2117, 18, 203imtr4i 291 . . . . . . 7 (βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯) β†’ βˆƒπ‘¦βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯))
2210, 21syl 17 . . . . . 6 (𝐴 β‰  βˆ… β†’ βˆƒπ‘¦βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯))
23 abn0 4381 . . . . . 6 ({𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} β‰  βˆ… ↔ βˆƒπ‘¦βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯))
2422, 23sylibr 233 . . . . 5 (𝐴 β‰  βˆ… β†’ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} β‰  βˆ…)
2511dfiin2 5037 . . . . . 6 ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = ∩ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)}
26 rankon 9818 . . . . . . . . . 10 (rankβ€˜π‘₯) ∈ On
27 eleq1 2813 . . . . . . . . . 10 (𝑦 = (rankβ€˜π‘₯) β†’ (𝑦 ∈ On ↔ (rankβ€˜π‘₯) ∈ On))
2826, 27mpbiri 257 . . . . . . . . 9 (𝑦 = (rankβ€˜π‘₯) β†’ 𝑦 ∈ On)
2928rexlimivw 3141 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯) β†’ 𝑦 ∈ On)
3029abssi 4064 . . . . . . 7 {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} βŠ† On
31 onint 7792 . . . . . . 7 (({𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} βŠ† On ∧ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} β‰  βˆ…) β†’ ∩ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)})
3230, 31mpan 688 . . . . . 6 ({𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} β‰  βˆ… β†’ ∩ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)})
3325, 32eqeltrid 2829 . . . . 5 ({𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} β‰  βˆ… β†’ ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)})
34 nfii1 5032 . . . . . . . . 9 β„²π‘₯∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯)
3534nfeq2 2910 . . . . . . . 8 β„²π‘₯ 𝑦 = ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯)
36 eqeq1 2729 . . . . . . . 8 (𝑦 = ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) β†’ (𝑦 = (rankβ€˜π‘₯) ↔ ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯)))
3735, 36rexbid 3262 . . . . . . 7 (𝑦 = ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) β†’ (βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯) ↔ βˆƒπ‘₯ ∈ 𝐴 ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯)))
3837elabg 3663 . . . . . 6 (∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} β†’ (∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} ↔ βˆƒπ‘₯ ∈ 𝐴 ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯)))
3938ibi 266 . . . . 5 (∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} β†’ βˆƒπ‘₯ ∈ 𝐴 ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯))
40 ssid 4000 . . . . . . . . . 10 (rankβ€˜π‘¦) βŠ† (rankβ€˜π‘¦)
41 fveq2 6894 . . . . . . . . . . . 12 (π‘₯ = 𝑦 β†’ (rankβ€˜π‘₯) = (rankβ€˜π‘¦))
4241sseq1d 4009 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ ((rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) ↔ (rankβ€˜π‘¦) βŠ† (rankβ€˜π‘¦)))
4342rspcev 3607 . . . . . . . . . 10 ((𝑦 ∈ 𝐴 ∧ (rankβ€˜π‘¦) βŠ† (rankβ€˜π‘¦)) β†’ βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))
4440, 43mpan2 689 . . . . . . . . 9 (𝑦 ∈ 𝐴 β†’ βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))
45 iinss 5059 . . . . . . . . 9 (βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) β†’ ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))
4644, 45syl 17 . . . . . . . 8 (𝑦 ∈ 𝐴 β†’ ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))
47 sseq1 4003 . . . . . . . 8 (∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯) β†’ (∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) ↔ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))
4846, 47imbitrid 243 . . . . . . 7 (∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯) β†’ (𝑦 ∈ 𝐴 β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))
4948ralrimiv 3135 . . . . . 6 (∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯) β†’ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))
5049reximi 3074 . . . . 5 (βˆƒπ‘₯ ∈ 𝐴 ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯) β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))
5124, 33, 39, 504syl 19 . . . 4 (𝐴 β‰  βˆ… β†’ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))
52 rabn0 4386 . . . 4 ({π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} β‰  βˆ… ↔ βˆƒπ‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))
5351, 52sylibr 233 . . 3 (𝐴 β‰  βˆ… β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} β‰  βˆ…)
5453necon4i 2966 . 2 ({π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} = βˆ… β†’ 𝐴 = βˆ…)
553, 54impbii 208 1 (𝐴 = βˆ… ↔ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} = βˆ…)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  {cab 2702   β‰  wne 2930  βˆ€wral 3051  βˆƒwrex 3060  {crab 3419   βŠ† wss 3945  βˆ…c0 4323  βˆ© cint 4949  βˆ© ciin 4997  Oncon0 6369  β€˜cfv 6547  rankcrnk 9786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  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 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7420  df-om 7870  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-r1 9787  df-rank 9788
This theorem is referenced by:  scott0s  9911  cplem1  9912  karden  9918  scott0f  37712  scotteld  43748
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