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Theorem scott0 9929
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 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem scott0
StepHypRef Expression
1 rabeq 3434 . . 3 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ ∅ ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
2 rab0 4387 . . 3 {𝑥 ∈ ∅ ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅
31, 2eqtrdi 2782 . 2 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
4 n0 4349 . . . . . . . 8 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 nfre1 3273 . . . . . . . . 9 𝑥𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)
6 eqid 2726 . . . . . . . . . 10 (rank‘𝑥) = (rank‘𝑥)
7 rspe 3237 . . . . . . . . . 10 ((𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
86, 7mpan2 689 . . . . . . . . 9 (𝑥𝐴 → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
95, 8exlimi 2206 . . . . . . . 8 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
104, 9sylbi 216 . . . . . . 7 (𝐴 ≠ ∅ → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
11 fvex 6914 . . . . . . . . . . 11 (rank‘𝑥) ∈ V
12 eqeq1 2730 . . . . . . . . . . . 12 (𝑦 = (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ (rank‘𝑥) = (rank‘𝑥)))
1312anbi2d 628 . . . . . . . . . . 11 (𝑦 = (rank‘𝑥) → ((𝑥𝐴𝑦 = (rank‘𝑥)) ↔ (𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥))))
1411, 13spcev 3592 . . . . . . . . . 10 ((𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
1514eximi 1830 . . . . . . . . 9 (∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
16 excom 2152 . . . . . . . . 9 (∃𝑦𝑥(𝑥𝐴𝑦 = (rank‘𝑥)) ↔ ∃𝑥𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
1715, 16sylibr 233 . . . . . . . 8 (∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
18 df-rex 3061 . . . . . . . 8 (∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) ↔ ∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)))
19 df-rex 3061 . . . . . . . . 9 (∃𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
2019exbii 1843 . . . . . . . 8 (∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑦𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
2117, 18, 203imtr4i 291 . . . . . . 7 (∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
2210, 21syl 17 . . . . . 6 (𝐴 ≠ ∅ → ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
23 abn0 4385 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ ↔ ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
2422, 23sylibr 233 . . . . 5 (𝐴 ≠ ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅)
2511dfiin2 5042 . . . . . 6 𝑥𝐴 (rank‘𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)}
26 rankon 9838 . . . . . . . . . 10 (rank‘𝑥) ∈ On
27 eleq1 2814 . . . . . . . . . 10 (𝑦 = (rank‘𝑥) → (𝑦 ∈ On ↔ (rank‘𝑥) ∈ On))
2826, 27mpbiri 257 . . . . . . . . 9 (𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
2928rexlimivw 3141 . . . . . . . 8 (∃𝑥𝐴 𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
3029abssi 4066 . . . . . . 7 {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ⊆ On
31 onint 7799 . . . . . . 7 (({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ⊆ On ∧ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
3230, 31mpan 688 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
3325, 32eqeltrid 2830 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
34 nfii1 5037 . . . . . . . . 9 𝑥 𝑥𝐴 (rank‘𝑥)
3534nfeq2 2910 . . . . . . . 8 𝑥 𝑦 = 𝑥𝐴 (rank‘𝑥)
36 eqeq1 2730 . . . . . . . 8 (𝑦 = 𝑥𝐴 (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3735, 36rexbid 3262 . . . . . . 7 (𝑦 = 𝑥𝐴 (rank‘𝑥) → (∃𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3837elabg 3664 . . . . . 6 ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} → ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ↔ ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3938ibi 266 . . . . 5 ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} → ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
40 ssid 4002 . . . . . . . . . 10 (rank‘𝑦) ⊆ (rank‘𝑦)
41 fveq2 6901 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
4241sseq1d 4011 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑦) ⊆ (rank‘𝑦)))
4342rspcev 3608 . . . . . . . . . 10 ((𝑦𝐴 ∧ (rank‘𝑦) ⊆ (rank‘𝑦)) → ∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
4440, 43mpan2 689 . . . . . . . . 9 (𝑦𝐴 → ∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
45 iinss 5064 . . . . . . . . 9 (∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
4644, 45syl 17 . . . . . . . 8 (𝑦𝐴 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
47 sseq1 4005 . . . . . . . 8 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ( 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
4846, 47imbitrid 243 . . . . . . 7 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → (𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
4948ralrimiv 3135 . . . . . 6 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
5049reximi 3074 . . . . 5 (∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ∃𝑥𝐴𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
5124, 33, 39, 504syl 19 . . . 4 (𝐴 ≠ ∅ → ∃𝑥𝐴𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
52 rabn0 4390 . . . 4 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑥𝐴𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
5351, 52sylibr 233 . . 3 (𝐴 ≠ ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)
5453necon4i 2966 . 2 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ → 𝐴 = ∅)
553, 54impbii 208 1 (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1534  wex 1774  wcel 2099  {cab 2703  wne 2930  wral 3051  wrex 3060  {crab 3419  wss 3947  c0 4325   cint 4954   ciin 5002  Oncon0 6376  cfv 6554  rankcrnk 9806
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 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-r1 9807  df-rank 9808
This theorem is referenced by:  scott0s  9931  cplem1  9932  karden  9938  scott0f  37870  scotteld  43920
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