Theorem wsuccl 36019

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 36004	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		wsuccl.1	. . . 4 ⊢ (𝜑 → 𝑅 We 𝐴)
3		weso 5615	. . . 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 36017	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
9	4, 8	infcl 9392	. 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴)
10	1, 9	eqeltrid 2840	1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2113  ∃wrex 3060   class class class wbr 5098   Or wor 5531   Se wse 5575   We wwe 5576  ^◡ccnv 5623  Predcpred 6258  infcinf 9344  wsuccwsuc 36002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-iota 6448  df-riota 7315  df-sup 9345  df-inf 9346  df-wsuc 36004

This theorem is referenced by: (None)