Theorem wsucex 33108

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	⊢ (𝜑 → 𝑅 Or 𝐴)

Assertion

Ref	Expression
wsucex	⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)

Proof of Theorem wsucex

Step	Hyp	Ref	Expression
1		df-wsuc 33094	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		wsucex.1	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐴)
3	2	infexd 8941	. 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ V)
4	1, 3	eqeltrid 2917	1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2110  Vcvv 3494   Or wor 5467  ^◡ccnv 5548  Predcpred 6141  infcinf 8899  wsuccwsuc 33092

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rmo 3146  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-po 5468  df-so 5469  df-cnv 5557  df-sup 8900  df-inf 8901  df-wsuc 33094

This theorem is referenced by: (None)