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Theorem scotteld 44270
Description: The Scott operation sends inhabited classes to inhabited sets. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypothesis
Ref Expression
scotteld.1 (𝜑 → ∃𝑥 𝑥𝐴)
Assertion
Ref Expression
scotteld (𝜑 → ∃𝑥 𝑥 ∈ Scott 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem scotteld
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scotteld.1 . . . . . 6 (𝜑 → ∃𝑥 𝑥𝐴)
2 n0 4352 . . . . . 6 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
31, 2sylibr 234 . . . . 5 (𝜑𝐴 ≠ ∅)
43neneqd 2944 . . . 4 (𝜑 → ¬ 𝐴 = ∅)
5 scott0 9927 . . . . 5 (𝐴 = ∅ ↔ {𝑦𝐴 ∣ ∀𝑧𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅)
6 df-scott 44260 . . . . . 6 Scott 𝐴 = {𝑦𝐴 ∣ ∀𝑧𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
76eqeq1i 2741 . . . . 5 (Scott 𝐴 = ∅ ↔ {𝑦𝐴 ∣ ∀𝑧𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅)
85, 7bitr4i 278 . . . 4 (𝐴 = ∅ ↔ Scott 𝐴 = ∅)
94, 8sylnib 328 . . 3 (𝜑 → ¬ Scott 𝐴 = ∅)
109neqned 2946 . 2 (𝜑 → Scott 𝐴 ≠ ∅)
11 n0 4352 . 2 (Scott 𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ Scott 𝐴)
1210, 11sylib 218 1 (𝜑 → ∃𝑥 𝑥 ∈ Scott 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wex 1778  wcel 2107  wne 2939  wral 3060  {crab 3435  wss 3950  c0 4332  cfv 6560  rankcrnk 9804  Scott cscott 44259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-r1 9805  df-rank 9806  df-scott 44260
This theorem is referenced by:  cpcolld  44282
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