Theorem swoord2 8046

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 22	. . . 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 3966	. . . . . . 7 ⊢ ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) ⊆ (𝑋 × 𝑋)
6	4, 5	eqsstri 3860	. . . . . 6 ⊢ 𝑅 ⊆ (𝑋 × 𝑋)
7	6	ssbri 4920	. . . . 5 ⊢ (𝐴𝑅𝐵 → 𝐴(𝑋 × 𝑋)𝐵)
8		df-br 4876	. . . . . 6 ⊢ (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
9		opelxp1 5387	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) → 𝐴 ∈ 𝑋)
10	8, 9	sylbi 209	. . . . 5 ⊢ (𝐴(𝑋 × 𝑋)𝐵 → 𝐴 ∈ 𝑋)
11	3, 7, 10	3syl 18	. . . 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 5274	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴)))
15	1, 2, 11, 12, 14	syl13anc 1495	. . 3 ⊢ (𝜑 → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴)))
16		idd 24	. . . 4 ⊢ (𝜑 → (𝐶 < 𝐵 → 𝐶 < 𝐵))
17	4	brdifun 8043	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
18	11, 12, 17	syl2anc 579	. . . . . . 7 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
19	3, 18	mpbid 224	. . . . . 6 ⊢ (𝜑 → ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
20		olc 899	. . . . . 6 ⊢ (𝐵 < 𝐴 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
21	19, 20	nsyl 138	. . . . 5 ⊢ (𝜑 → ¬ 𝐵 < 𝐴)
22	21	pm2.21d 119	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐴 → 𝐶 < 𝐵))
23	16, 22	jaod 890	. . 3 ⊢ (𝜑 → ((𝐶 < 𝐵 ∨ 𝐵 < 𝐴) → 𝐶 < 𝐵))
24	15, 23	syld 47	. 2 ⊢ (𝜑 → (𝐶 < 𝐴 → 𝐶 < 𝐵))
25	13	swopolem 5274	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐶 < 𝐵 → (𝐶 < 𝐴 ∨ 𝐴 < 𝐵)))
26	1, 2, 12, 11, 25	syl13anc 1495	. . 3 ⊢ (𝜑 → (𝐶 < 𝐵 → (𝐶 < 𝐴 ∨ 𝐴 < 𝐵)))
27		idd 24	. . . 4 ⊢ (𝜑 → (𝐶 < 𝐴 → 𝐶 < 𝐴))
28		orc 898	. . . . . 6 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
29	19, 28	nsyl 138	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 < 𝐵)
30	29	pm2.21d 119	. . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐶 < 𝐴))
31	27, 30	jaod 890	. . 3 ⊢ (𝜑 → ((𝐶 < 𝐴 ∨ 𝐴 < 𝐵) → 𝐶 < 𝐴))
32	26, 31	syld 47	. 2 ⊢ (𝜑 → (𝐶 < 𝐵 → 𝐶 < 𝐴))
33	24, 32	impbid 204	1 ⊢ (𝜑 → (𝐶 < 𝐴 ↔ 𝐶 < 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 878   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ∖ cdif 3795   ∪ cun 3796  ⟨cop 4405   class class class wbr 4875   × cxp 5344  ^◡ccnv 5345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-cnv 5354

This theorem is referenced by: (None)