ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  swoord2 Structured version   GIF version

Theorem swoord2 6036
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 (y 𝑋 z 𝑋)) → (y < z → ¬ z < y))
swoer.3 ((φ (x 𝑋 y 𝑋 z 𝑋)) → (x < y → (x < z z < y)))
swoord.4 (φB 𝑋)
swoord.5 (φ𝐶 𝑋)
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 19 . . . 4 (φφ)
2 swoord.5 . . . 4 (φ𝐶 𝑋)
3 swoord.6 . . . . 5 (φA𝑅B)
4 swoer.1 . . . . . . 7 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
5 difss 3039 . . . . . . 7 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
64, 5eqsstri 2944 . . . . . 6 𝑅 ⊆ (𝑋 × 𝑋)
76ssbri 3770 . . . . 5 (A𝑅BA(𝑋 × 𝑋)B)
8 df-br 3729 . . . . . 6 (A(𝑋 × 𝑋)B ↔ ⟨A, B (𝑋 × 𝑋))
9 opelxp1 4293 . . . . . 6 (⟨A, B (𝑋 × 𝑋) → A 𝑋)
108, 9sylbi 114 . . . . 5 (A(𝑋 × 𝑋)BA 𝑋)
113, 7, 103syl 17 . . . 4 (φA 𝑋)
12 swoord.4 . . . 4 (φB 𝑋)
13 swoer.3 . . . . 5 ((φ (x 𝑋 y 𝑋 z 𝑋)) → (x < y → (x < z z < y)))
1413swopolem 4006 . . . 4 ((φ (𝐶 𝑋 A 𝑋 B 𝑋)) → (𝐶 < A → (𝐶 < B B < A)))
151, 2, 11, 12, 14syl13anc 1118 . . 3 (φ → (𝐶 < A → (𝐶 < B B < A)))
16 idd 21 . . . 4 (φ → (𝐶 < B𝐶 < B))
174brdifun 6033 . . . . . . . 8 ((A 𝑋 B 𝑋) → (A𝑅B ↔ ¬ (A < B B < A)))
1811, 12, 17syl2anc 391 . . . . . . 7 (φ → (A𝑅B ↔ ¬ (A < B B < A)))
193, 18mpbid 135 . . . . . 6 (φ → ¬ (A < B B < A))
20 olc 616 . . . . . 6 (B < A → (A < B B < A))
2119, 20nsyl 543 . . . . 5 (φ → ¬ B < A)
2221pm2.21d 534 . . . 4 (φ → (B < A𝐶 < B))
2316, 22jaod 621 . . 3 (φ → ((𝐶 < B B < A) → 𝐶 < B))
2415, 23syld 40 . 2 (φ → (𝐶 < A𝐶 < B))
2513swopolem 4006 . . . 4 ((φ (𝐶 𝑋 B 𝑋 A 𝑋)) → (𝐶 < B → (𝐶 < A A < B)))
261, 2, 12, 11, 25syl13anc 1118 . . 3 (φ → (𝐶 < B → (𝐶 < A A < B)))
27 idd 21 . . . 4 (φ → (𝐶 < A𝐶 < A))
28 orc 617 . . . . . 6 (A < B → (A < B B < A))
2919, 28nsyl 543 . . . . 5 (φ → ¬ A < B)
3029pm2.21d 534 . . . 4 (φ → (A < B𝐶 < A))
3127, 30jaod 621 . . 3 (φ → ((𝐶 < A A < B) → 𝐶 < A))
3226, 31syld 40 . 2 (φ → (𝐶 < B𝐶 < A))
3324, 32impbid 120 1 (φ → (𝐶 < A𝐶 < B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 613   w3a 867   = wceq 1224   wcel 1367  cdif 2883  cun 2884  cop 3343   class class class wbr 3728   × cxp 4259  ccnv 4260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 529  ax-in2 530  ax-io 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-v 2529  df-dif 2889  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-br 3729  df-opab 3783  df-xp 4267  df-cnv 4269
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator