Theorem wsucex 33820

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 33806	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		wsucex.1	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐴)
3	2	infexd 9242	. 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ V)
4	1, 3	eqeltrid 2843	1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3432   Or wor 5502  ^◡ccnv 5588  Predcpred 6201  infcinf 9200  wsuccwsuc 33804

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-po 5503  df-so 5504  df-cnv 5597  df-sup 9201  df-inf 9202  df-wsuc 33806

This theorem is referenced by: (None)