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Theorem wsucex 34440
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 34426 . 2 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2 wsucex.1 . . 3 (𝜑𝑅 Or 𝐴)
32infexd 9426 . 2 (𝜑 → inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ V)
41, 3eqeltrid 2842 1 (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3448   Or wor 5549  ccnv 5637  Predcpred 6257  infcinf 9384  wsuccwsuc 34424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-po 5550  df-so 5551  df-cnv 5646  df-sup 9385  df-inf 9386  df-wsuc 34426
This theorem is referenced by: (None)
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