Theorem wsuccl 36804

Description: If 𝑋 is a set with an 𝑅 successor in 𝐴, then its well-founded successor is a member of 𝐴. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)

Hypotheses

Ref	Expression
wsuccl.1	⊢ (𝜑 → 𝑅 We 𝐴)
wsuccl.2	⊢ (𝜑 → 𝑅 Se 𝐴)
wsuccl.3	⊢ (𝜑 → 𝑋 ∈ 𝑉)
wsuccl.4	⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦)

Assertion

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

Distinct variable groups:   𝑦,𝑅   𝑦,𝐴   𝑦,𝑋

Allowed substitution hints:   𝜑(𝑦)   𝑉(𝑦)

Proof of Theorem wsuccl

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wsuc 36789	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		wsuccl.1	. . . 4 ⊢ (𝜑 → 𝑅 We 𝐴)
3		weso 5649	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
4	2, 3	syl 18	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐴)
5		wsuccl.2	. . . 4 ⊢ (𝜑 → 𝑅 Se 𝐴)
6		wsuccl.3	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
7		wsuccl.4	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦)
8	2, 5, 6, 7	wsuclem 36802	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
9	4, 8	infcl 9518	. 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴)
10	1, 9	eqeltrid 2868	1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ∃wrex 3090   class class class wbr 5108   Or wor 5565   Se wse 5609   We wwe 5610  ^◡ccnv 5657  Predcpred 6329  infcinf 9470  wsuccwsuc 36787

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-iota 6520  df-riota 7408  df-sup 9471  df-inf 9472  df-wsuc 36789

This theorem is used by: (None)