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Theorem wsucex 33436
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 33422 . 2 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2 wsucex.1 . . 3 (𝜑𝑅 Or 𝐴)
32infexd 9023 . 2 (𝜑 → inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ V)
41, 3eqeltrid 2838 1 (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  Vcvv 3399   Or wor 5442  ccnv 5525  Predcpred 6129  infcinf 8981  wsuccwsuc 33420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rmo 3062  df-rab 3063  df-v 3401  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-po 5443  df-so 5444  df-cnv 5534  df-sup 8982  df-inf 8983  df-wsuc 33422
This theorem is referenced by: (None)
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