Theorem wsuccl 36021

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 36006	. 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2		wsuccl.1	. . . 4 ⊢ (𝜑 → 𝑅 We 𝐴)
3		weso 5613	. . . 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 36019	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
9	4, 8	infcl 9393	. 2 ⊢ (𝜑 → inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴)
10	1, 9	eqeltrid 2841	1 ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2114  ∃wrex 3062   class class class wbr 5086   Or wor 5529   Se wse 5573   We wwe 5574  ^◡ccnv 5621  Predcpred 6256  infcinf 9345  wsuccwsuc 36004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-iota 6446  df-riota 7315  df-sup 9346  df-inf 9347  df-wsuc 36006

This theorem is referenced by: (None)