MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  swoord2 Structured version   Visualization version   GIF version

Theorem swoord2 7726
Description: The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
Hypotheses
Ref Expression
swoer.1 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
swoer.2 ((𝜑 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))
swoer.3 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
swoord.4 (𝜑𝐵𝑋)
swoord.5 (𝜑𝐶𝑋)
swoord.6 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
swoord2 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
Distinct variable groups:   𝑥,𝑦,𝑧, <   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧
Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧)

Proof of Theorem swoord2
StepHypRef Expression
1 id 22 . . . 4 (𝜑𝜑)
2 swoord.5 . . . 4 (𝜑𝐶𝑋)
3 swoord.6 . . . . 5 (𝜑𝐴𝑅𝐵)
4 swoer.1 . . . . . . 7 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
5 difss 3720 . . . . . . 7 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
64, 5eqsstri 3619 . . . . . 6 𝑅 ⊆ (𝑋 × 𝑋)
76ssbri 4662 . . . . 5 (𝐴𝑅𝐵𝐴(𝑋 × 𝑋)𝐵)
8 df-br 4619 . . . . . 6 (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
9 opelxp1 5115 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴𝑋)
108, 9sylbi 207 . . . . 5 (𝐴(𝑋 × 𝑋)𝐵𝐴𝑋)
113, 7, 103syl 18 . . . 4 (𝜑𝐴𝑋)
12 swoord.4 . . . 4 (𝜑𝐵𝑋)
13 swoer.3 . . . . 5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))
1413swopolem 5009 . . . 4 ((𝜑 ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐶 < 𝐴 → (𝐶 < 𝐵𝐵 < 𝐴)))
151, 2, 11, 12, 14syl13anc 1325 . . 3 (𝜑 → (𝐶 < 𝐴 → (𝐶 < 𝐵𝐵 < 𝐴)))
16 idd 24 . . . 4 (𝜑 → (𝐶 < 𝐵𝐶 < 𝐵))
174brdifun 7723 . . . . . . . 8 ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
1811, 12, 17syl2anc 692 . . . . . . 7 (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
193, 18mpbid 222 . . . . . 6 (𝜑 → ¬ (𝐴 < 𝐵𝐵 < 𝐴))
20 olc 399 . . . . . 6 (𝐵 < 𝐴 → (𝐴 < 𝐵𝐵 < 𝐴))
2119, 20nsyl 135 . . . . 5 (𝜑 → ¬ 𝐵 < 𝐴)
2221pm2.21d 118 . . . 4 (𝜑 → (𝐵 < 𝐴𝐶 < 𝐵))
2316, 22jaod 395 . . 3 (𝜑 → ((𝐶 < 𝐵𝐵 < 𝐴) → 𝐶 < 𝐵))
2415, 23syld 47 . 2 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
2513swopolem 5009 . . . 4 ((𝜑 ∧ (𝐶𝑋𝐵𝑋𝐴𝑋)) → (𝐶 < 𝐵 → (𝐶 < 𝐴𝐴 < 𝐵)))
261, 2, 12, 11, 25syl13anc 1325 . . 3 (𝜑 → (𝐶 < 𝐵 → (𝐶 < 𝐴𝐴 < 𝐵)))
27 idd 24 . . . 4 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐴))
28 orc 400 . . . . . 6 (𝐴 < 𝐵 → (𝐴 < 𝐵𝐵 < 𝐴))
2919, 28nsyl 135 . . . . 5 (𝜑 → ¬ 𝐴 < 𝐵)
3029pm2.21d 118 . . . 4 (𝜑 → (𝐴 < 𝐵𝐶 < 𝐴))
3127, 30jaod 395 . . 3 (𝜑 → ((𝐶 < 𝐴𝐴 < 𝐵) → 𝐶 < 𝐴))
3226, 31syld 47 . 2 (𝜑 → (𝐶 < 𝐵𝐶 < 𝐴))
3324, 32impbid 202 1 (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  cdif 3556  cun 3557  cop 4159   class class class wbr 4618   × cxp 5077  ccnv 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator