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Theorem wsucex 35827
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 35813 . 2 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2 wsucex.1 . . 3 (𝜑𝑅 Or 𝐴)
32infexd 9523 . 2 (𝜑 → inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ V)
41, 3eqeltrid 2845 1 (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  Vcvv 3480   Or wor 5591  ccnv 5684  Predcpred 6320  infcinf 9481  wsuccwsuc 35811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-po 5592  df-so 5593  df-cnv 5693  df-sup 9482  df-inf 9483  df-wsuc 35813
This theorem is referenced by: (None)
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