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Theorem swoord1 6071
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
swoord1 (φ → (A < 𝐶B < 𝐶))
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 swoord1
StepHypRef Expression
1 id 19 . . . 4 (φφ)
2 swoord.6 . . . . 5 (φA𝑅B)
3 swoer.1 . . . . . . 7 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
4 difss 3064 . . . . . . 7 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
53, 4eqsstri 2969 . . . . . 6 𝑅 ⊆ (𝑋 × 𝑋)
65ssbri 3797 . . . . 5 (A𝑅BA(𝑋 × 𝑋)B)
7 df-br 3756 . . . . . 6 (A(𝑋 × 𝑋)B ↔ ⟨A, B (𝑋 × 𝑋))
8 opelxp1 4320 . . . . . 6 (⟨A, B (𝑋 × 𝑋) → A 𝑋)
97, 8sylbi 114 . . . . 5 (A(𝑋 × 𝑋)BA 𝑋)
102, 6, 93syl 17 . . . 4 (φA 𝑋)
11 swoord.5 . . . 4 (φ𝐶 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 4033 . . . 4 ((φ (A 𝑋 𝐶 𝑋 B 𝑋)) → (A < 𝐶 → (A < B B < 𝐶)))
151, 10, 11, 12, 14syl13anc 1136 . . 3 (φ → (A < 𝐶 → (A < B B < 𝐶)))
163brdifun 6069 . . . . . . 7 ((A 𝑋 B 𝑋) → (A𝑅B ↔ ¬ (A < B B < A)))
1710, 12, 16syl2anc 391 . . . . . 6 (φ → (A𝑅B ↔ ¬ (A < B B < A)))
182, 17mpbid 135 . . . . 5 (φ → ¬ (A < B B < A))
19 orc 632 . . . . 5 (A < B → (A < B B < A))
2018, 19nsyl 558 . . . 4 (φ → ¬ A < B)
21 biorf 662 . . . 4 A < B → (B < 𝐶 ↔ (A < B B < 𝐶)))
2220, 21syl 14 . . 3 (φ → (B < 𝐶 ↔ (A < B B < 𝐶)))
2315, 22sylibrd 158 . 2 (φ → (A < 𝐶B < 𝐶))
2413swopolem 4033 . . . 4 ((φ (B 𝑋 𝐶 𝑋 A 𝑋)) → (B < 𝐶 → (B < A A < 𝐶)))
251, 12, 11, 10, 24syl13anc 1136 . . 3 (φ → (B < 𝐶 → (B < A A < 𝐶)))
26 olc 631 . . . . 5 (B < A → (A < B B < A))
2718, 26nsyl 558 . . . 4 (φ → ¬ B < A)
28 biorf 662 . . . 4 B < A → (A < 𝐶 ↔ (B < A A < 𝐶)))
2927, 28syl 14 . . 3 (φ → (A < 𝐶 ↔ (B < A A < 𝐶)))
3025, 29sylibrd 158 . 2 (φ → (B < 𝐶A < 𝐶))
3123, 30impbid 120 1 (φ → (A < 𝐶B < 𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   w3a 884   = wceq 1242   wcel 1390  cdif 2908  cun 2909  cop 3370   class class class wbr 3755   × cxp 4286  ccnv 4287
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296
This theorem is referenced by: (None)
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