Theorem swoso 8736

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 3931	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
8	1	sselda 3931	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
9	7, 8	anim12dan 631	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
10		swoer.1	. . . . . . 7 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))
11	10	brdifun 8732	. . . . . 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 1105	. . . . . . 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 607	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦)
16	15	expr 462	. . . . 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 877	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
19		3orcomb 1110	. . . 4 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥 ∨ 𝑥 = 𝑦))
20		df-3or 1104	. . . 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)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899   class class class wbr 5103   Po wpo 5557   Or wor 5558   × cxp 5649  ^◡ccnv 5650

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-po 5559  df-so 5560  df-xp 5657  df-cnv 5659

This theorem is used by: (None)