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Theorem wsuclb 35810
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 5893 . . . . . 6 ((𝑌𝐴𝑋𝑉) → (𝑌𝑅𝑋𝑋𝑅𝑌))
52, 3, 4syl2anc 584 . . . . 5 (𝜑 → (𝑌𝑅𝑋𝑋𝑅𝑌))
61, 5mpbird 257 . . . 4 (𝜑𝑌𝑅𝑋)
7 elpredg 6337 . . . . 5 ((𝑋𝑉𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
83, 2, 7syl2anc 584 . . . 4 (𝜑 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
96, 8mpbird 257 . . 3 (𝜑𝑌 ∈ Pred(𝑅, 𝐴, 𝑋))
10 wsuclb.1 . . . . 5 (𝜑𝑅 We 𝐴)
11 weso 5680 . . . . 5 (𝑅 We 𝐴𝑅 Or 𝐴)
1210, 11syl 17 . . . 4 (𝜑𝑅 Or 𝐴)
13 wsuclb.2 . . . . 5 (𝜑𝑅 Se 𝐴)
14 breq2 5152 . . . . . . 7 (𝑦 = 𝑌 → (𝑋𝑅𝑦𝑋𝑅𝑌))
1514rspcev 3622 . . . . . 6 ((𝑌𝐴𝑋𝑅𝑌) → ∃𝑦𝐴 𝑋𝑅𝑦)
162, 1, 15syl2anc 584 . . . . 5 (𝜑 → ∃𝑦𝐴 𝑋𝑅𝑦)
1710, 13, 3, 16wsuclem 35807 . . . 4 (𝜑 → ∃𝑎𝐴 (∀𝑏 ∈ Pred (𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
1812, 17inflb 9527 . . 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 35794 . . 3 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2120breq2i 5156 . 2 (𝑌𝑅wsuc(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))
2219, 21sylnibr 329 1 (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wcel 2106  wrex 3068   class class class wbr 5148   Or wor 5596   Se wse 5639   We wwe 5640  ccnv 5688  Predcpred 6322  infcinf 9479  wsuccwsuc 35792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-riota 7388  df-sup 9480  df-inf 9481  df-wsuc 35794
This theorem is referenced by: (None)
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