Theorem swoso 8680

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 5551	. . 3 ⊢ (𝜑 → < Po 𝑋)
5		poss 5542	. . 3 ⊢ (𝑌 ⊆ 𝑋 → ( < Po 𝑋 → < Po 𝑌))
6	1, 4, 5	sylc 65	. 2 ⊢ (𝜑 → < Po 𝑌)
7	1	sselda 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
8	1	sselda 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
9	7, 8	anim12dan 620	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
10		swoer.1	. . . . . . 7 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))
11	10	brdifun 8676	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥)))
12	9, 11	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥)))
13		df-3an 1089	. . . . . . 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 596	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦)
16	15	expr 456	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑅𝑦 → 𝑥 = 𝑦))
17	12, 16	sylbird 260	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (¬ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥) → 𝑥 = 𝑦))
18	17	orrd 864	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
19		3orcomb 1094	. . . 4 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥 ∨ 𝑥 = 𝑦))
20		df-3or 1088	. . . 4 ⊢ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥 ∨ 𝑥 = 𝑦) ↔ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
21	19, 20	bitri 275	. . 3 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
22	18, 21	sylibr 234	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
23	6, 22	issod 5575	1 ⊢ (𝜑 → < Or 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∨ w3o 1086   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903   class class class wbr 5100   Po wpo 5538   Or wor 5539   × cxp 5630  ^◡ccnv 5631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-po 5540  df-so 5541  df-xp 5638  df-cnv 5640

This theorem is referenced by: (None)