Theorem swoso 8732

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 5598	. . 3 ⊢ (𝜑 → < Po 𝑋)
5		poss 5589	. . 3 ⊢ (𝑌 ⊆ 𝑋 → ( < Po 𝑋 → < Po 𝑌))
6	1, 4, 5	sylc 65	. 2 ⊢ (𝜑 → < Po 𝑌)
7	1	sselda 3981	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
8	1	sselda 3981	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
9	7, 8	anim12dan 619	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
10		swoer.1	. . . . . . 7 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))
11	10	brdifun 8728	. . . . . 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 595	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦)
16	15	expr 457	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑅𝑦 → 𝑥 = 𝑦))
17	12, 16	sylbird 259	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (¬ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥) → 𝑥 = 𝑦))
18	17	orrd 861	. . 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 274	. . 3 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
22	18, 21	sylibr 233	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
23	6, 22	issod 5620	1 ⊢ (𝜑 → < Or 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947   class class class wbr 5147   Po wpo 5585   Or wor 5586   × cxp 5673  ^◡ccnv 5674

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-po 5587  df-so 5588  df-xp 5681  df-cnv 5683

This theorem is referenced by: (None)