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Theorem scott0 9924
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 3448 . . 3 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ ∅ ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
2 rab0 4392 . . 3 {𝑥 ∈ ∅ ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅
31, 2eqtrdi 2791 . 2 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
4 n0 4359 . . . . . . . 8 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 nfre1 3283 . . . . . . . . 9 𝑥𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)
6 eqid 2735 . . . . . . . . . 10 (rank‘𝑥) = (rank‘𝑥)
7 rspe 3247 . . . . . . . . . 10 ((𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
86, 7mpan2 691 . . . . . . . . 9 (𝑥𝐴 → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
95, 8exlimi 2215 . . . . . . . 8 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
104, 9sylbi 217 . . . . . . 7 (𝐴 ≠ ∅ → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
11 fvex 6920 . . . . . . . . . . 11 (rank‘𝑥) ∈ V
12 eqeq1 2739 . . . . . . . . . . . 12 (𝑦 = (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ (rank‘𝑥) = (rank‘𝑥)))
1312anbi2d 630 . . . . . . . . . . 11 (𝑦 = (rank‘𝑥) → ((𝑥𝐴𝑦 = (rank‘𝑥)) ↔ (𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥))))
1411, 13spcev 3606 . . . . . . . . . 10 ((𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
1514eximi 1832 . . . . . . . . 9 (∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
16 excom 2160 . . . . . . . . 9 (∃𝑦𝑥(𝑥𝐴𝑦 = (rank‘𝑥)) ↔ ∃𝑥𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
1715, 16sylibr 234 . . . . . . . 8 (∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑦𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
18 df-rex 3069 . . . . . . . 8 (∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) ↔ ∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)))
19 df-rex 3069 . . . . . . . . 9 (∃𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
2019exbii 1845 . . . . . . . 8 (∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑦𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
2117, 18, 203imtr4i 292 . . . . . . 7 (∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
2210, 21syl 17 . . . . . 6 (𝐴 ≠ ∅ → ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
23 abn0 4391 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ ↔ ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
2422, 23sylibr 234 . . . . 5 (𝐴 ≠ ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅)
2511dfiin2 5039 . . . . . 6 𝑥𝐴 (rank‘𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)}
26 rankon 9833 . . . . . . . . . 10 (rank‘𝑥) ∈ On
27 eleq1 2827 . . . . . . . . . 10 (𝑦 = (rank‘𝑥) → (𝑦 ∈ On ↔ (rank‘𝑥) ∈ On))
2826, 27mpbiri 258 . . . . . . . . 9 (𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
2928rexlimivw 3149 . . . . . . . 8 (∃𝑥𝐴 𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
3029abssi 4080 . . . . . . 7 {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ⊆ On
31 onint 7810 . . . . . . 7 (({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ⊆ On ∧ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
3230, 31mpan 690 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
3325, 32eqeltrid 2843 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
34 nfii1 5034 . . . . . . . . 9 𝑥 𝑥𝐴 (rank‘𝑥)
3534nfeq2 2921 . . . . . . . 8 𝑥 𝑦 = 𝑥𝐴 (rank‘𝑥)
36 eqeq1 2739 . . . . . . . 8 (𝑦 = 𝑥𝐴 (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3735, 36rexbid 3272 . . . . . . 7 (𝑦 = 𝑥𝐴 (rank‘𝑥) → (∃𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3837elabg 3677 . . . . . 6 ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} → ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ↔ ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3938ibi 267 . . . . 5 ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} → ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
40 ssid 4018 . . . . . . . . . 10 (rank‘𝑦) ⊆ (rank‘𝑦)
41 fveq2 6907 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
4241sseq1d 4027 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑦) ⊆ (rank‘𝑦)))
4342rspcev 3622 . . . . . . . . . 10 ((𝑦𝐴 ∧ (rank‘𝑦) ⊆ (rank‘𝑦)) → ∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
4440, 43mpan2 691 . . . . . . . . 9 (𝑦𝐴 → ∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
45 iinss 5061 . . . . . . . . 9 (∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
4644, 45syl 17 . . . . . . . 8 (𝑦𝐴 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
47 sseq1 4021 . . . . . . . 8 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ( 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
4846, 47imbitrid 244 . . . . . . 7 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → (𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
4948ralrimiv 3143 . . . . . 6 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
5049reximi 3082 . . . . 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 4395 . . . 4 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑥𝐴𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
5351, 52sylibr 234 . . 3 (𝐴 ≠ ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)
5453necon4i 2974 . 2 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ → 𝐴 = ∅)
553, 54impbii 209 1 (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  {crab 3433  wss 3963  c0 4339   cint 4951   ciin 4997  Oncon0 6386  cfv 6563  rankcrnk 9801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803
This theorem is referenced by:  scott0s  9926  cplem1  9927  karden  9933  scott0f  38156  scotteld  44242
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