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Theorem wsuccl 32361
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 32346 . 2 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2 wsuccl.1 . . . 4 (𝜑𝑅 We 𝐴)
3 weso 5346 . . . 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 32359 . . 3 (𝜑 → ∃𝑎𝐴 (∀𝑏 ∈ Pred (𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
94, 8infcl 8682 . 2 (𝜑 → inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅) ∈ 𝐴)
101, 9syl5eqel 2863 1 (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wrex 3091   class class class wbr 4886   Or wor 5273   Se wse 5312   We wwe 5313  ccnv 5354  Predcpred 5932  infcinf 8635  wsuccwsuc 32344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-iota 6099  df-riota 6883  df-sup 8636  df-inf 8637  df-wsuc 32346
This theorem is referenced by: (None)
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