Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wsuclb Structured version   Visualization version   GIF version

Theorem wsuclb 34442
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 5840 . . . . . 6 ((𝑌𝐴𝑋𝑉) → (𝑌𝑅𝑋𝑋𝑅𝑌))
52, 3, 4syl2anc 585 . . . . 5 (𝜑 → (𝑌𝑅𝑋𝑋𝑅𝑌))
61, 5mpbird 257 . . . 4 (𝜑𝑌𝑅𝑋)
7 elpredg 6272 . . . . 5 ((𝑋𝑉𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
83, 2, 7syl2anc 585 . . . 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 5629 . . . . 5 (𝑅 We 𝐴𝑅 Or 𝐴)
1210, 11syl 17 . . . 4 (𝜑𝑅 Or 𝐴)
13 wsuclb.2 . . . . 5 (𝜑𝑅 Se 𝐴)
14 breq2 5114 . . . . . . 7 (𝑦 = 𝑌 → (𝑋𝑅𝑦𝑋𝑅𝑌))
1514rspcev 3584 . . . . . 6 ((𝑌𝐴𝑋𝑅𝑌) → ∃𝑦𝐴 𝑋𝑅𝑦)
162, 1, 15syl2anc 585 . . . . 5 (𝜑 → ∃𝑦𝐴 𝑋𝑅𝑦)
1710, 13, 3, 16wsuclem 34439 . . . 4 (𝜑 → ∃𝑎𝐴 (∀𝑏 ∈ Pred (𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
1812, 17inflb 9432 . . 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 34426 . . 3 wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
2120breq2i 5118 . 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 205  wcel 2107  wrex 3074   class class class wbr 5110   Or wor 5549   Se wse 5591   We wwe 5592  ccnv 5637  Predcpred 6257  infcinf 9384  wsuccwsuc 34424
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-iota 6453  df-riota 7318  df-sup 9385  df-inf 9386  df-wsuc 34426
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator