Theorem wsucex 36510

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 36496	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		wsucex.1	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐴)
3	2	infexd 9454	. 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ V)
4	1, 3	eqeltrid 2864	1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3450   Or wor 5554  ^◡ccnv 5646  Predcpred 6292  infcinf 9411  wsuccwsuc 36494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-po 5555  df-so 5556  df-cnv 5655  df-sup 9412  df-inf 9413  df-wsuc 36496

This theorem is used by: (None)