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Theorem wsuccl 31897
 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 (𝜑𝑅 We 𝐴)
wsuccl.2 (𝜑𝑅 Se 𝐴)
wsuccl.3 (𝜑𝑋𝑉)
wsuccl.4 (𝜑 → ∃𝑦𝐴 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.
StepHypRef Expression
1 df-wsuc 31882 . 2 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2 wsuccl.1 . . . 4 (𝜑𝑅 We 𝐴)
3 weso 5134 . . . 4 (𝑅 We 𝐴𝑅 Or 𝐴)
42, 3syl 17 . . 3 (𝜑𝑅 Or 𝐴)
5 wsuccl.2 . . . 4 (𝜑𝑅 Se 𝐴)
6 wsuccl.3 . . . 4 (𝜑𝑋𝑉)
7 wsuccl.4 . . . 4 (𝜑 → ∃𝑦𝐴 𝑋𝑅𝑦)
82, 5, 6, 7wsuclem 31895 . . 3 (𝜑 → ∃𝑎𝐴 (∀𝑏 ∈ Pred (𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
94, 8infcl 8435 . 2 (𝜑 → inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴)
101, 9syl5eqel 2734 1 (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2030  ∃wrex 2942   class class class wbr 4685   Or wor 5063   Se wse 5100   We wwe 5101  ◡ccnv 5142  Predcpred 5717  infcinf 8388  wsuccwsuc 31880 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-iota 5889  df-riota 6651  df-sup 8389  df-inf 8390  df-wsuc 31882 This theorem is referenced by: (None)
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