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Theorem swoord1 6035
 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 3039 . . . . . . 7 ((𝑋 × 𝑋) ∖ ( < < )) ⊆ (𝑋 × 𝑋)
53, 4eqsstri 2944 . . . . . 6 𝑅 ⊆ (𝑋 × 𝑋)
65ssbri 3770 . . . . 5 (A𝑅BA(𝑋 × 𝑋)B)
7 df-br 3729 . . . . . 6 (A(𝑋 × 𝑋)B ↔ ⟨A, B (𝑋 × 𝑋))
8 opelxp1 4293 . . . . . 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 4006 . . . 4 ((φ (A 𝑋 𝐶 𝑋 B 𝑋)) → (A < 𝐶 → (A < B B < 𝐶)))
151, 10, 11, 12, 14syl13anc 1118 . . 3 (φ → (A < 𝐶 → (A < B B < 𝐶)))
163brdifun 6033 . . . . . . 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 617 . . . . 5 (A < B → (A < B B < A))
2018, 19nsyl 543 . . . 4 (φ → ¬ A < B)
21 biorf 647 . . . 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 4006 . . . 4 ((φ (B 𝑋 𝐶 𝑋 A 𝑋)) → (B < 𝐶 → (B < A A < 𝐶)))
251, 12, 11, 10, 24syl13anc 1118 . . 3 (φ → (B < 𝐶 → (B < A A < 𝐶)))
26 olc 616 . . . . 5 (B < A → (A < B B < A))
2718, 26nsyl 543 . . . 4 (φ → ¬ B < A)
28 biorf 647 . . . 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 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)
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