Theorem wsuccl 35803

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 35788	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		wsuccl.1	. . . 4 ⊢ (𝜑 → 𝑅 We 𝐴)
3		weso 5614	. . . 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 35801	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
9	4, 8	infcl 9398	. 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴)
10	1, 9	eqeltrid 2832	1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2109  ∃wrex 3053   class class class wbr 5095   Or wor 5530   Se wse 5574   We wwe 5575  ^◡ccnv 5622  Predcpred 6252  infcinf 9350  wsuccwsuc 35786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-iota 6442  df-riota 7310  df-sup 9351  df-inf 9352  df-wsuc 35788

This theorem is referenced by: (None)