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Theorem wsuclb 35555
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 5882 . . . . . 6 ((𝑌𝐴𝑋𝑉) → (𝑌𝑅𝑋𝑋𝑅𝑌))
52, 3, 4syl2anc 582 . . . . 5 (𝜑 → (𝑌𝑅𝑋𝑋𝑅𝑌))
61, 5mpbird 256 . . . 4 (𝜑𝑌𝑅𝑋)
7 elpredg 6321 . . . . 5 ((𝑋𝑉𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
83, 2, 7syl2anc 582 . . . 4 (𝜑 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
96, 8mpbird 256 . . 3 (𝜑𝑌 ∈ Pred(𝑅, 𝐴, 𝑋))
10 wsuclb.1 . . . . 5 (𝜑𝑅 We 𝐴)
11 weso 5669 . . . . 5 (𝑅 We 𝐴𝑅 Or 𝐴)
1210, 11syl 17 . . . 4 (𝜑𝑅 Or 𝐴)
13 wsuclb.2 . . . . 5 (𝜑𝑅 Se 𝐴)
14 breq2 5153 . . . . . . 7 (𝑦 = 𝑌 → (𝑋𝑅𝑦𝑋𝑅𝑌))
1514rspcev 3606 . . . . . 6 ((𝑌𝐴𝑋𝑅𝑌) → ∃𝑦𝐴 𝑋𝑅𝑦)
162, 1, 15syl2anc 582 . . . . 5 (𝜑 → ∃𝑦𝐴 𝑋𝑅𝑦)
1710, 13, 3, 16wsuclem 35552 . . . 4 (𝜑 → ∃𝑎𝐴 (∀𝑏 ∈ Pred (𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
1812, 17inflb 9514 . . 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 35539 . . 3 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2120breq2i 5157 . 2 (𝑌𝑅wsuc(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))
2219, 21sylnibr 328 1 (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wcel 2098  wrex 3059   class class class wbr 5149   Or wor 5589   Se wse 5631   We wwe 5632  ccnv 5677  Predcpred 6306  infcinf 9466  wsuccwsuc 35537
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 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-iota 6501  df-riota 7375  df-sup 9467  df-inf 9468  df-wsuc 35539
This theorem is referenced by: (None)
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