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Theorem wsuccl 35322
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 35307 . 2 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2 wsuccl.1 . . . 4 (𝜑𝑅 We 𝐴)
3 weso 5658 . . . 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 35320 . . 3 (𝜑 → ∃𝑎𝐴 (∀𝑏 ∈ Pred (𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
94, 8infcl 9480 . 2 (𝜑 → inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴)
101, 9eqeltrid 2829 1 (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wrex 3062   class class class wbr 5139   Or wor 5578   Se wse 5620   We wwe 5621  ccnv 5666  Predcpred 6290  infcinf 9433  wsuccwsuc 35305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-iota 6486  df-riota 7358  df-sup 9434  df-inf 9435  df-wsuc 35307
This theorem is referenced by: (None)
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