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Theorem scott0 9901
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 3441 . . 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 4378 . . 3 {π‘₯ ∈ βˆ… ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} = βˆ…
31, 2eqtrdi 2783 . 2 (𝐴 = βˆ… β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} = βˆ…)
4 n0 4342 . . . . . . . 8 (𝐴 β‰  βˆ… ↔ βˆƒπ‘₯ π‘₯ ∈ 𝐴)
5 nfre1 3277 . . . . . . . . 9 β„²π‘₯βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯)
6 eqid 2727 . . . . . . . . . 10 (rankβ€˜π‘₯) = (rankβ€˜π‘₯)
7 rspe 3241 . . . . . . . . . 10 ((π‘₯ ∈ 𝐴 ∧ (rankβ€˜π‘₯) = (rankβ€˜π‘₯)) β†’ βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯))
86, 7mpan2 690 . . . . . . . . 9 (π‘₯ ∈ 𝐴 β†’ βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯))
95, 8exlimi 2203 . . . . . . . 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 6904 . . . . . . . . . . 11 (rankβ€˜π‘₯) ∈ V
12 eqeq1 2731 . . . . . . . . . . . 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 1830 . . . . . . . . 9 (βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ (rankβ€˜π‘₯) = (rankβ€˜π‘₯)) β†’ βˆƒπ‘₯βˆƒπ‘¦(π‘₯ ∈ 𝐴 ∧ 𝑦 = (rankβ€˜π‘₯)))
16 excom 2155 . . . . . . . . 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 3066 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯) ↔ βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ (rankβ€˜π‘₯) = (rankβ€˜π‘₯)))
19 df-rex 3066 . . . . . . . . 9 (βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯) ↔ βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ 𝑦 = (rankβ€˜π‘₯)))
2019exbii 1843 . . . . . . . 8 (βˆƒπ‘¦βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯) ↔ βˆƒπ‘¦βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ 𝑦 = (rankβ€˜π‘₯)))
2117, 18, 203imtr4i 292 . . . . . . 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 4376 . . . . . 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 5031 . . . . . 6 ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = ∩ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)}
26 rankon 9810 . . . . . . . . . 10 (rankβ€˜π‘₯) ∈ On
27 eleq1 2816 . . . . . . . . . 10 (𝑦 = (rankβ€˜π‘₯) β†’ (𝑦 ∈ On ↔ (rankβ€˜π‘₯) ∈ On))
2826, 27mpbiri 258 . . . . . . . . 9 (𝑦 = (rankβ€˜π‘₯) β†’ 𝑦 ∈ On)
2928rexlimivw 3146 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯) β†’ 𝑦 ∈ On)
3029abssi 4063 . . . . . . 7 {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} βŠ† On
31 onint 7787 . . . . . . 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 689 . . . . . 6 ({𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} β‰  βˆ… β†’ ∩ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)})
3325, 32eqeltrid 2832 . . . . 5 ({𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)} β‰  βˆ… β†’ ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = (rankβ€˜π‘₯)})
34 nfii1 5026 . . . . . . . . 9 β„²π‘₯∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯)
3534nfeq2 2915 . . . . . . . 8 β„²π‘₯ 𝑦 = ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯)
36 eqeq1 2731 . . . . . . . 8 (𝑦 = ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) β†’ (𝑦 = (rankβ€˜π‘₯) ↔ ∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯)))
3735, 36rexbid 3266 . . . . . . 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 267 . . . . 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 6891 . . . . . . . . . . . 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 690 . . . . . . . . 9 (𝑦 ∈ 𝐴 β†’ βˆƒπ‘₯ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))
45 iinss 5053 . . . . . . . . 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 3140 . . . . . 6 (∩ π‘₯ ∈ 𝐴 (rankβ€˜π‘₯) = (rankβ€˜π‘₯) β†’ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))
5049reximi 3079 . . . . 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 4381 . . . 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 2971 . 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 395   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099  {cab 2704   β‰  wne 2935  βˆ€wral 3056  βˆƒwrex 3065  {crab 3427   βŠ† wss 3944  βˆ…c0 4318  βˆ© cint 4944  βˆ© ciin 4992  Oncon0 6363  β€˜cfv 6542  rankcrnk 9778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7417  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-r1 9779  df-rank 9780
This theorem is referenced by:  scott0s  9903  cplem1  9904  karden  9910  scott0f  37577  scotteld  43606
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