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Theorem wsucex 31019
 Description: Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
Hypothesis
Ref Expression
wsucex.1 (𝜑𝑅 Or 𝐴)
Assertion
Ref Expression
wsucex (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)

Proof of Theorem wsucex
StepHypRef Expression
1 df-wsuc 31000 . 2 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2 wsucex.1 . . 3 (𝜑𝑅 Or 𝐴)
32infexd 8272 . 2 (𝜑 → inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ V)
41, 3syl5eqel 2692 1 (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  Vcvv 3173   Or wor 4958  ◡ccnv 5037  Predcpred 5596  infcinf 8230  wsuccwsuc 30996 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rmo 2904  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-cnv 5046  df-sup 8231  df-inf 8232  df-wsuc 31000 This theorem is referenced by: (None)
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