Theorem swoord2 6560

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	⊢ (𝜑 → 𝐵 ∈ 𝑋)
swoord.5	⊢ (𝜑 → 𝐶 ∈ 𝑋)
swoord.6	⊢ (𝜑 → 𝐴𝑅𝐵)

Assertion

Ref	Expression
swoord2	⊢ (𝜑 → (𝐶 < 𝐴 ↔ 𝐶 < 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

Step	Hyp	Ref	Expression
1		id 19	. . . 4 ⊢ (𝜑 → 𝜑)
2		swoord.5	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
3		swoord.6	. . . . 5 ⊢ (𝜑 → 𝐴𝑅𝐵)
4		swoer.1	. . . . . . 7 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))
5		difss 3261	. . . . . . 7 ⊢ ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) ⊆ (𝑋 × 𝑋)
6	4, 5	eqsstri 3187	. . . . . 6 ⊢ 𝑅 ⊆ (𝑋 × 𝑋)
7	6	ssbri 4045	. . . . 5 ⊢ (𝐴𝑅𝐵 → 𝐴(𝑋 × 𝑋)𝐵)
8		df-br 4002	. . . . . 6 ⊢ (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
9		opelxp1 4658	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴 ∈ 𝑋)
10	8, 9	sylbi 121	. . . . 5 ⊢ (𝐴(𝑋 × 𝑋)𝐵 → 𝐴 ∈ 𝑋)
11	3, 7, 10	3syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
12		swoord.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑋)
13		swoer.3	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
14	13	swopolem 4303	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴)))
15	1, 2, 11, 12, 14	syl13anc 1240	. . 3 ⊢ (𝜑 → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴)))
16		idd 21	. . . 4 ⊢ (𝜑 → (𝐶 < 𝐵 → 𝐶 < 𝐵))
17	4	brdifun 6557	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
18	11, 12, 17	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
19	3, 18	mpbid 147	. . . . . 6 ⊢ (𝜑 → ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
20		olc 711	. . . . . 6 ⊢ (𝐵 < 𝐴 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
21	19, 20	nsyl 628	. . . . 5 ⊢ (𝜑 → ¬ 𝐵 < 𝐴)
22	21	pm2.21d 619	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐴 → 𝐶 < 𝐵))
23	16, 22	jaod 717	. . 3 ⊢ (𝜑 → ((𝐶 < 𝐵 ∨ 𝐵 < 𝐴) → 𝐶 < 𝐵))
24	15, 23	syld 45	. 2 ⊢ (𝜑 → (𝐶 < 𝐴 → 𝐶 < 𝐵))
25	13	swopolem 4303	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐶 < 𝐵 → (𝐶 < 𝐴 ∨ 𝐴 < 𝐵)))
26	1, 2, 12, 11, 25	syl13anc 1240	. . 3 ⊢ (𝜑 → (𝐶 < 𝐵 → (𝐶 < 𝐴 ∨ 𝐴 < 𝐵)))
27		idd 21	. . . 4 ⊢ (𝜑 → (𝐶 < 𝐴 → 𝐶 < 𝐴))
28		orc 712	. . . . . 6 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
29	19, 28	nsyl 628	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 < 𝐵)
30	29	pm2.21d 619	. . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐶 < 𝐴))
31	27, 30	jaod 717	. . 3 ⊢ (𝜑 → ((𝐶 < 𝐴 ∨ 𝐴 < 𝐵) → 𝐶 < 𝐴))
32	26, 31	syld 45	. 2 ⊢ (𝜑 → (𝐶 < 𝐵 → 𝐶 < 𝐴))
33	24, 32	impbid 129	1 ⊢ (𝜑 → (𝐶 < 𝐴 ↔ 𝐶 < 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   ∧ w3a 978   = wceq 1353   ∈ wcel 2148   ∖ cdif 3126   ∪ cun 3127  ⟨cop 3595   class class class wbr 4001   × cxp 4622  ^◡ccnv 4623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4002  df-opab 4063  df-xp 4630  df-cnv 4632

This theorem is referenced by: (None)