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Theorem scotteld 40874
 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 4293 . . . . . 6 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
31, 2sylibr 237 . . . . 5 (𝜑𝐴 ≠ ∅)
43neneqd 3019 . . . 4 (𝜑 → ¬ 𝐴 = ∅)
5 scott0 9312 . . . . 5 (𝐴 = ∅ ↔ {𝑦𝐴 ∣ ∀𝑧𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅)
6 df-scott 40864 . . . . . 6 Scott 𝐴 = {𝑦𝐴 ∣ ∀𝑧𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
76eqeq1i 2829 . . . . 5 (Scott 𝐴 = ∅ ↔ {𝑦𝐴 ∣ ∀𝑧𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅)
85, 7bitr4i 281 . . . 4 (𝐴 = ∅ ↔ Scott 𝐴 = ∅)
94, 8sylnib 331 . . 3 (𝜑 → ¬ Scott 𝐴 = ∅)
109neqned 3021 . 2 (𝜑 → Scott 𝐴 ≠ ∅)
11 n0 4293 . 2 (Scott 𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ Scott 𝐴)
1210, 11sylib 221 1 (𝜑 → ∃𝑥 𝑥 ∈ Scott 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  ∃wex 1781   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  {crab 3137   ⊆ wss 3919  ∅c0 4276  ‘cfv 6343  rankcrnk 9189  Scott cscott 40863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-r1 9190  df-rank 9191  df-scott 40864 This theorem is referenced by:  cpcolld  40886
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