Theorem wsucex 33117

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 33103	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		wsucex.1	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐴)
3	2	infexd 8950	. 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ V)
4	1, 3	eqeltrid 2920	1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2113  Vcvv 3497   Or wor 5476  ^◡ccnv 5557  Predcpred 6150  infcinf 8908  wsuccwsuc 33101

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rmo 3149  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-po 5477  df-so 5478  df-cnv 5566  df-sup 8909  df-inf 8910  df-wsuc 33103

This theorem is referenced by: (None)