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Theorem wsuclb 33377
 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 5720 . . . . . 6 ((𝑌𝐴𝑋𝑉) → (𝑌𝑅𝑋𝑋𝑅𝑌))
52, 3, 4syl2anc 588 . . . . 5 (𝜑 → (𝑌𝑅𝑋𝑋𝑅𝑌))
61, 5mpbird 260 . . . 4 (𝜑𝑌𝑅𝑋)
7 elpredg 6141 . . . . 5 ((𝑋𝑉𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
83, 2, 7syl2anc 588 . . . 4 (𝜑 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
96, 8mpbird 260 . . 3 (𝜑𝑌 ∈ Pred(𝑅, 𝐴, 𝑋))
10 wsuclb.1 . . . . 5 (𝜑𝑅 We 𝐴)
11 weso 5516 . . . . 5 (𝑅 We 𝐴𝑅 Or 𝐴)
1210, 11syl 17 . . . 4 (𝜑𝑅 Or 𝐴)
13 wsuclb.2 . . . . 5 (𝜑𝑅 Se 𝐴)
14 breq2 5037 . . . . . . 7 (𝑦 = 𝑌 → (𝑋𝑅𝑦𝑋𝑅𝑌))
1514rspcev 3542 . . . . . 6 ((𝑌𝐴𝑋𝑅𝑌) → ∃𝑦𝐴 𝑋𝑅𝑦)
162, 1, 15syl2anc 588 . . . . 5 (𝜑 → ∃𝑦𝐴 𝑋𝑅𝑦)
1710, 13, 3, 16wsuclem 33374 . . . 4 (𝜑 → ∃𝑎𝐴 (∀𝑏 ∈ Pred (𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
1812, 17inflb 8987 . . 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 33361 . . 3 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2120breq2i 5041 . 2 (𝑌𝑅wsuc(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))
2219, 21sylnibr 333 1 (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∈ wcel 2112  ∃wrex 3072   class class class wbr 5033   Or wor 5443   Se wse 5482   We wwe 5483  ◡ccnv 5524  Predcpred 6126  infcinf 8939  wsuccwsuc 33359 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-po 5444  df-so 5445  df-fr 5484  df-se 5485  df-we 5486  df-xp 5531  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-iota 6295  df-riota 7109  df-sup 8940  df-inf 8941  df-wsuc 33361 This theorem is referenced by: (None)
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