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Theorem wsuclb 31898
Description: A well-founded successor is a lower bound on points after 𝑋. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
Hypotheses
Ref Expression
wsuclb.1 (𝜑𝑅 We 𝐴)
wsuclb.2 (𝜑𝑅 Se 𝐴)
wsuclb.3 (𝜑𝑋𝑉)
wsuclb.4 (𝜑𝑌𝐴)
wsuclb.5 (𝜑𝑋𝑅𝑌)
Assertion
Ref Expression
wsuclb (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))

Proof of Theorem wsuclb
Dummy variables 𝑎 𝑏 𝑐 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wsuclb.5 . . . . 5 (𝜑𝑋𝑅𝑌)
2 wsuclb.4 . . . . . 6 (𝜑𝑌𝐴)
3 wsuclb.3 . . . . . 6 (𝜑𝑋𝑉)
4 brcnvg 5335 . . . . . 6 ((𝑌𝐴𝑋𝑉) → (𝑌𝑅𝑋𝑋𝑅𝑌))
52, 3, 4syl2anc 694 . . . . 5 (𝜑 → (𝑌𝑅𝑋𝑋𝑅𝑌))
61, 5mpbird 247 . . . 4 (𝜑𝑌𝑅𝑋)
7 elpredg 5732 . . . . 5 ((𝑋𝑉𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
83, 2, 7syl2anc 694 . . . 4 (𝜑 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
96, 8mpbird 247 . . 3 (𝜑𝑌 ∈ Pred(𝑅, 𝐴, 𝑋))
10 wsuclb.1 . . . . 5 (𝜑𝑅 We 𝐴)
11 weso 5134 . . . . 5 (𝑅 We 𝐴𝑅 Or 𝐴)
1210, 11syl 17 . . . 4 (𝜑𝑅 Or 𝐴)
13 wsuclb.2 . . . . 5 (𝜑𝑅 Se 𝐴)
14 breq2 4689 . . . . . . 7 (𝑦 = 𝑌 → (𝑋𝑅𝑦𝑋𝑅𝑌))
1514rspcev 3340 . . . . . 6 ((𝑌𝐴𝑋𝑅𝑌) → ∃𝑦𝐴 𝑋𝑅𝑦)
162, 1, 15syl2anc 694 . . . . 5 (𝜑 → ∃𝑦𝐴 𝑋𝑅𝑦)
1710, 13, 3, 16wsuclem 31895 . . . 4 (𝜑 → ∃𝑎𝐴 (∀𝑏 ∈ Pred (𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
1812, 17inflb 8436 . . 3 (𝜑 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → ¬ 𝑌𝑅inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))
199, 18mpd 15 . 2 (𝜑 → ¬ 𝑌𝑅inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))
20 df-wsuc 31882 . . 3 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2120breq2i 4693 . 2 (𝑌𝑅wsuc(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))
2219, 21sylnibr 318 1 (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  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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