Theorem swoer 8776

Description: Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

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)))

Assertion

Ref	Expression
swoer	⊢ (𝜑 → 𝑅 Er 𝑋)

Distinct variable groups:   𝑥,𝑦,𝑧, <   𝜑,𝑥,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧)

Proof of Theorem swoer

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		swoer.1	. . . . 5 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))
2		difss 4136	. . . . 5 ⊢ ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) ⊆ (𝑋 × 𝑋)
3	1, 2	eqsstri 4030	. . . 4 ⊢ 𝑅 ⊆ (𝑋 × 𝑋)
4		relxp 5703	. . . 4 ⊢ Rel (𝑋 × 𝑋)
5		relss 5791	. . . 4 ⊢ (𝑅 ⊆ (𝑋 × 𝑋) → (Rel (𝑋 × 𝑋) → Rel 𝑅))
6	3, 4, 5	mp2 9	. . 3 ⊢ Rel 𝑅
7	6	a1i 11	. 2 ⊢ (𝜑 → Rel 𝑅)
8		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑢𝑅𝑣) → 𝑢𝑅𝑣)
9		orcom 871	. . . . . 6 ⊢ ((𝑢 < 𝑣 ∨ 𝑣 < 𝑢) ↔ (𝑣 < 𝑢 ∨ 𝑢 < 𝑣))
10	9	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑢𝑅𝑣) → ((𝑢 < 𝑣 ∨ 𝑣 < 𝑢) ↔ (𝑣 < 𝑢 ∨ 𝑢 < 𝑣)))
11	10	notbid 318	. . . 4 ⊢ ((𝜑 ∧ 𝑢𝑅𝑣) → (¬ (𝑢 < 𝑣 ∨ 𝑣 < 𝑢) ↔ ¬ (𝑣 < 𝑢 ∨ 𝑢 < 𝑣)))
12	3	ssbri 5188	. . . . . . 7 ⊢ (𝑢𝑅𝑣 → 𝑢(𝑋 × 𝑋)𝑣)
13	12	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢𝑅𝑣) → 𝑢(𝑋 × 𝑋)𝑣)
14		brxp 5734	. . . . . 6 ⊢ (𝑢(𝑋 × 𝑋)𝑣 ↔ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋))
15	13, 14	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑢𝑅𝑣) → (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋))
16	1	brdifun 8775	. . . . 5 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣 ∨ 𝑣 < 𝑢)))
17	15, 16	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑢𝑅𝑣) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣 ∨ 𝑣 < 𝑢)))
18	15	simprd 495	. . . . 5 ⊢ ((𝜑 ∧ 𝑢𝑅𝑣) → 𝑣 ∈ 𝑋)
19	15	simpld 494	. . . . 5 ⊢ ((𝜑 ∧ 𝑢𝑅𝑣) → 𝑢 ∈ 𝑋)
20	1	brdifun 8775	. . . . 5 ⊢ ((𝑣 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋) → (𝑣𝑅𝑢 ↔ ¬ (𝑣 < 𝑢 ∨ 𝑢 < 𝑣)))
21	18, 19, 20	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑢𝑅𝑣) → (𝑣𝑅𝑢 ↔ ¬ (𝑣 < 𝑢 ∨ 𝑢 < 𝑣)))
22	11, 17, 21	3bitr4d 311	. . 3 ⊢ ((𝜑 ∧ 𝑢𝑅𝑣) → (𝑢𝑅𝑣 ↔ 𝑣𝑅𝑢))
23	8, 22	mpbid 232	. 2 ⊢ ((𝜑 ∧ 𝑢𝑅𝑣) → 𝑣𝑅𝑢)
24		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → 𝑢𝑅𝑣)
25	12	ad2antrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → 𝑢(𝑋 × 𝑋)𝑣)
26	14	simplbi 497	. . . . . . 7 ⊢ (𝑢(𝑋 × 𝑋)𝑣 → 𝑢 ∈ 𝑋)
27	25, 26	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → 𝑢 ∈ 𝑋)
28	14	simprbi 496	. . . . . . 7 ⊢ (𝑢(𝑋 × 𝑋)𝑣 → 𝑣 ∈ 𝑋)
29	25, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → 𝑣 ∈ 𝑋)
30	27, 29, 16	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → (𝑢𝑅𝑣 ↔ ¬ (𝑢 < 𝑣 ∨ 𝑣 < 𝑢)))
31	24, 30	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → ¬ (𝑢 < 𝑣 ∨ 𝑣 < 𝑢))
32		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → 𝑣𝑅𝑤)
33	3	brel 5750	. . . . . . . 8 ⊢ (𝑣𝑅𝑤 → (𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋))
34	33	simprd 495	. . . . . . 7 ⊢ (𝑣𝑅𝑤 → 𝑤 ∈ 𝑋)
35	32, 34	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → 𝑤 ∈ 𝑋)
36	1	brdifun 8775	. . . . . 6 ⊢ ((𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑣𝑅𝑤 ↔ ¬ (𝑣 < 𝑤 ∨ 𝑤 < 𝑣)))
37	29, 35, 36	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → (𝑣𝑅𝑤 ↔ ¬ (𝑣 < 𝑤 ∨ 𝑤 < 𝑣)))
38	32, 37	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → ¬ (𝑣 < 𝑤 ∨ 𝑤 < 𝑣))
39		simpl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → 𝜑)
40		swoer.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
41	40	swopolem 5602	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢 < 𝑤 → (𝑢 < 𝑣 ∨ 𝑣 < 𝑤)))
42	39, 27, 35, 29, 41	syl13anc 1374	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → (𝑢 < 𝑤 → (𝑢 < 𝑣 ∨ 𝑣 < 𝑤)))
43	40	swopolem 5602	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑤 < 𝑢 → (𝑤 < 𝑣 ∨ 𝑣 < 𝑢)))
44	39, 35, 27, 29, 43	syl13anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → (𝑤 < 𝑢 → (𝑤 < 𝑣 ∨ 𝑣 < 𝑢)))
45		orcom 871	. . . . . . 7 ⊢ ((𝑣 < 𝑢 ∨ 𝑤 < 𝑣) ↔ (𝑤 < 𝑣 ∨ 𝑣 < 𝑢))
46	44, 45	imbitrrdi 252	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → (𝑤 < 𝑢 → (𝑣 < 𝑢 ∨ 𝑤 < 𝑣)))
47	42, 46	orim12d 967	. . . . 5 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → ((𝑢 < 𝑤 ∨ 𝑤 < 𝑢) → ((𝑢 < 𝑣 ∨ 𝑣 < 𝑤) ∨ (𝑣 < 𝑢 ∨ 𝑤 < 𝑣))))
48		or4 927	. . . . 5 ⊢ (((𝑢 < 𝑣 ∨ 𝑣 < 𝑤) ∨ (𝑣 < 𝑢 ∨ 𝑤 < 𝑣)) ↔ ((𝑢 < 𝑣 ∨ 𝑣 < 𝑢) ∨ (𝑣 < 𝑤 ∨ 𝑤 < 𝑣)))
49	47, 48	imbitrdi 251	. . . 4 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → ((𝑢 < 𝑤 ∨ 𝑤 < 𝑢) → ((𝑢 < 𝑣 ∨ 𝑣 < 𝑢) ∨ (𝑣 < 𝑤 ∨ 𝑤 < 𝑣))))
50	31, 38, 49	mtord 880	. . 3 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → ¬ (𝑢 < 𝑤 ∨ 𝑤 < 𝑢))
51	1	brdifun 8775	. . . 4 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑢𝑅𝑤 ↔ ¬ (𝑢 < 𝑤 ∨ 𝑤 < 𝑢)))
52	27, 35, 51	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → (𝑢𝑅𝑤 ↔ ¬ (𝑢 < 𝑤 ∨ 𝑤 < 𝑢)))
53	50, 52	mpbird 257	. 2 ⊢ ((𝜑 ∧ (𝑢𝑅𝑣 ∧ 𝑣𝑅𝑤)) → 𝑢𝑅𝑤)
54		swoer.2	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))
55	54, 40	swopo 5603	. . . . . 6 ⊢ (𝜑 → < Po 𝑋)
56		poirr 5604	. . . . . 6 ⊢ (( < Po 𝑋 ∧ 𝑢 ∈ 𝑋) → ¬ 𝑢 < 𝑢)
57	55, 56	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑋) → ¬ 𝑢 < 𝑢)
58		pm1.2 904	. . . . 5 ⊢ ((𝑢 < 𝑢 ∨ 𝑢 < 𝑢) → 𝑢 < 𝑢)
59	57, 58	nsyl 140	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑋) → ¬ (𝑢 < 𝑢 ∨ 𝑢 < 𝑢))
60		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑋) → 𝑢 ∈ 𝑋)
61	1	brdifun 8775	. . . . 5 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋) → (𝑢𝑅𝑢 ↔ ¬ (𝑢 < 𝑢 ∨ 𝑢 < 𝑢)))
62	60, 60, 61	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑋) → (𝑢𝑅𝑢 ↔ ¬ (𝑢 < 𝑢 ∨ 𝑢 < 𝑢)))
63	59, 62	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑋) → 𝑢𝑅𝑢)
64	3	ssbri 5188	. . . . 5 ⊢ (𝑢𝑅𝑢 → 𝑢(𝑋 × 𝑋)𝑢)
65		brxp 5734	. . . . . 6 ⊢ (𝑢(𝑋 × 𝑋)𝑢 ↔ (𝑢 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋))
66	65	simplbi 497	. . . . 5 ⊢ (𝑢(𝑋 × 𝑋)𝑢 → 𝑢 ∈ 𝑋)
67	64, 66	syl 17	. . . 4 ⊢ (𝑢𝑅𝑢 → 𝑢 ∈ 𝑋)
68	67	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑢𝑅𝑢) → 𝑢 ∈ 𝑋)
69	63, 68	impbida 801	. 2 ⊢ (𝜑 → (𝑢 ∈ 𝑋 ↔ 𝑢𝑅𝑢))
70	7, 23, 53, 69	iserd 8771	1 ⊢ (𝜑 → 𝑅 Er 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ∖ cdif 3948   ∪ cun 3949   ⊆ wss 3951   class class class wbr 5143   Po wpo 5590   × cxp 5683  ^◡ccnv 5684  Rel wrel 5690   Er wer 8742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-po 5592  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-er 8745

This theorem is referenced by: (None)