Theorem wsuccl 35791

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 35776	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		wsuccl.1	. . . 4 ⊢ (𝜑 → 𝑅 We 𝐴)
3		weso 5691	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
4	2, 3	syl 17	. . 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 35789	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
9	4, 8	infcl 9557	. 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴)
10	1, 9	eqeltrid 2848	1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2108  ∃wrex 3076   class class class wbr 5166   Or wor 5606   Se wse 5650   We wwe 5651  ^◡ccnv 5699  Predcpred 6331  infcinf 9510  wsuccwsuc 35774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-riota 7404  df-sup 9511  df-inf 9512  df-wsuc 35776

This theorem is referenced by: (None)