Users' Mathboxes Mathbox for Rohan Ridenour < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  scotteld Structured version   Visualization version   GIF version

Theorem scotteld 44242
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 4359 . . . . . 6 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
31, 2sylibr 234 . . . . 5 (𝜑𝐴 ≠ ∅)
43neneqd 2943 . . . 4 (𝜑 → ¬ 𝐴 = ∅)
5 scott0 9924 . . . . 5 (𝐴 = ∅ ↔ {𝑦𝐴 ∣ ∀𝑧𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅)
6 df-scott 44232 . . . . . 6 Scott 𝐴 = {𝑦𝐴 ∣ ∀𝑧𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
76eqeq1i 2740 . . . . 5 (Scott 𝐴 = ∅ ↔ {𝑦𝐴 ∣ ∀𝑧𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅)
85, 7bitr4i 278 . . . 4 (𝐴 = ∅ ↔ Scott 𝐴 = ∅)
94, 8sylnib 328 . . 3 (𝜑 → ¬ Scott 𝐴 = ∅)
109neqned 2945 . 2 (𝜑 → Scott 𝐴 ≠ ∅)
11 n0 4359 . 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 1537  wex 1776  wcel 2106  wne 2938  wral 3059  {crab 3433  wss 3963  c0 4339  cfv 6563  rankcrnk 9801  Scott cscott 44231
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  df-scott 44232
This theorem is referenced by:  cpcolld  44254
  Copyright terms: Public domain W3C validator