Theorem swoso 8717

Description: If the incomparability relation is equivalent to equality in a subset, then the partial order strictly orders the subset. (Contributed by Mario Carneiro, 30-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)))
swoso.4	⊢ (𝜑 → 𝑌 ⊆ 𝑋)
swoso.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦)

Assertion

Ref	Expression
swoso	⊢ (𝜑 → < Or 𝑌)

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

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

Proof of Theorem swoso

Step	Hyp	Ref	Expression
1		swoso.4	. . 3 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
2		swoer.2	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))
3		swoer.3	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧 ∨ 𝑧 < 𝑦)))
4	2, 3	swopo 5570	. . 3 ⊢ (𝜑 → < Po 𝑋)
5		poss 5561	. . 3 ⊢ (𝑌 ⊆ 𝑋 → ( < Po 𝑋 → < Po 𝑌))
6	1, 4, 5	sylc 66	. 2 ⊢ (𝜑 → < Po 𝑌)
7	1	sselda 3939	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
8	1	sselda 3939	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
9	7, 8	anim12dan 630	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
10		swoer.1	. . . . . . 7 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))
11	10	brdifun 8713	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥)))
12	9, 11	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥)))
13		df-3an 1103	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ∧ 𝑥𝑅𝑦) ↔ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ 𝑥𝑅𝑦))
14		swoso.5	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦)
15	13, 14	sylan2br 606	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦)
16	15	expr 461	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑅𝑦 → 𝑥 = 𝑦))
17	12, 16	sylbird 263	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (¬ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥) → 𝑥 = 𝑦))
18	17	orrd 876	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
19		3orcomb 1108	. . . 4 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥 ∨ 𝑥 = 𝑦))
20		df-3or 1102	. . . 4 ⊢ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥 ∨ 𝑥 = 𝑦) ↔ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
21	19, 20	bitri 278	. . 3 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
22	18, 21	sylibr 237	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
23	6, 22	issod 5594	1 ⊢ (𝜑 → < Or 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∨ w3o 1100   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ∖ cdif 3904   ∪ cun 3905   ⊆ wss 3907   class class class wbr 5104   Po wpo 5557   Or wor 5558   × cxp 5649  ^◡ccnv 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-po 5559  df-so 5560  df-xp 5657  df-cnv 5659

This theorem is referenced by: (None)