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Theorem scott0 9926
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 3451 . . 3 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ ∅ ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
2 rab0 4386 . . 3 {𝑥 ∈ ∅ ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅
31, 2eqtrdi 2793 . 2 (𝐴 = ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
4 n0 4353 . . . . . . . 8 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
5 nfre1 3285 . . . . . . . . 9 𝑥𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)
6 eqid 2737 . . . . . . . . . 10 (rank‘𝑥) = (rank‘𝑥)
7 rspe 3249 . . . . . . . . . 10 ((𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
86, 7mpan2 691 . . . . . . . . 9 (𝑥𝐴 → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
95, 8exlimi 2217 . . . . . . . 8 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
104, 9sylbi 217 . . . . . . 7 (𝐴 ≠ ∅ → ∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥))
11 fvex 6919 . . . . . . . . . . 11 (rank‘𝑥) ∈ V
12 eqeq1 2741 . . . . . . . . . . . 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 1835 . . . . . . . . 9 (∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)) → ∃𝑥𝑦(𝑥𝐴𝑦 = (rank‘𝑥)))
16 excom 2162 . . . . . . . . 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 3071 . . . . . . . 8 (∃𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) ↔ ∃𝑥(𝑥𝐴 ∧ (rank‘𝑥) = (rank‘𝑥)))
19 df-rex 3071 . . . . . . . . 9 (∃𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥(𝑥𝐴𝑦 = (rank‘𝑥)))
2019exbii 1848 . . . . . . . 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 4385 . . . . . 6 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ ↔ ∃𝑦𝑥𝐴 𝑦 = (rank‘𝑥))
2422, 23sylibr 234 . . . . 5 (𝐴 ≠ ∅ → {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅)
2511dfiin2 5034 . . . . . 6 𝑥𝐴 (rank‘𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)}
26 rankon 9835 . . . . . . . . . 10 (rank‘𝑥) ∈ On
27 eleq1 2829 . . . . . . . . . 10 (𝑦 = (rank‘𝑥) → (𝑦 ∈ On ↔ (rank‘𝑥) ∈ On))
2826, 27mpbiri 258 . . . . . . . . 9 (𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
2928rexlimivw 3151 . . . . . . . 8 (∃𝑥𝐴 𝑦 = (rank‘𝑥) → 𝑦 ∈ On)
3029abssi 4070 . . . . . . 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 2845 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)} ≠ ∅ → 𝑥𝐴 (rank‘𝑥) ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (rank‘𝑥)})
34 nfii1 5029 . . . . . . . . 9 𝑥 𝑥𝐴 (rank‘𝑥)
3534nfeq2 2923 . . . . . . . 8 𝑥 𝑦 = 𝑥𝐴 (rank‘𝑥)
36 eqeq1 2741 . . . . . . . 8 (𝑦 = 𝑥𝐴 (rank‘𝑥) → (𝑦 = (rank‘𝑥) ↔ 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3735, 36rexbid 3274 . . . . . . 7 (𝑦 = 𝑥𝐴 (rank‘𝑥) → (∃𝑥𝐴 𝑦 = (rank‘𝑥) ↔ ∃𝑥𝐴 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥)))
3837elabg 3676 . . . . . 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 4006 . . . . . . . . . 10 (rank‘𝑦) ⊆ (rank‘𝑦)
41 fveq2 6906 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
4241sseq1d 4015 . . . . . . . . . . 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 5056 . . . . . . . . 9 (∃𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
4644, 45syl 17 . . . . . . . 8 (𝑦𝐴 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
47 sseq1 4009 . . . . . . . 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 3145 . . . . . 6 ( 𝑥𝐴 (rank‘𝑥) = (rank‘𝑥) → ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
5049reximi 3084 . . . . 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 4389 . . . 4 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑥𝐴𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦))
5351, 52sylibr 234 . . 3 (𝐴 ≠ ∅ → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)
5453necon4i 2976 . 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 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  {crab 3436  wss 3951  c0 4333   cint 4946   ciin 4992  Oncon0 6384  cfv 6561  rankcrnk 9803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805
This theorem is referenced by:  scott0s  9928  cplem1  9929  karden  9935  scott0f  38176  scotteld  44265
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