Theorem wsucex 36187

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 36173	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		wsucex.1	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐴)
3	2	infexd 9432	. 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ V)
4	1, 3	eqeltrid 2869	1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2145  Vcvv 3457   Or wor 5559  ^◡ccnv 5651  Predcpred 6291  infcinf 9389  wsuccwsuc 36171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-po 5560  df-so 5561  df-cnv 5660  df-sup 9390  df-inf 9391  df-wsuc 36173

This theorem is referenced by: (None)