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Mirrors > Home > MPE Home > Th. List > swoso | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
swoso | ⊢ (𝜑 → < Or 𝑌) |
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 5484 | . . 3 ⊢ (𝜑 → < Po 𝑋) |
5 | poss 5476 | . . 3 ⊢ (𝑌 ⊆ 𝑋 → ( < Po 𝑋 → < Po 𝑌)) | |
6 | 1, 4, 5 | sylc 65 | . 2 ⊢ (𝜑 → < Po 𝑌) |
7 | 1 | sselda 3967 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋) |
8 | 1 | sselda 3967 | . . . . . . 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 8318 | . . . . . 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 1085 | . . . . . . 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 459 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑅𝑦 → 𝑥 = 𝑦)) |
17 | 12, 16 | sylbird 262 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (¬ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥) → 𝑥 = 𝑦)) |
18 | 17 | orrd 859 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦)) |
19 | 3orcomb 1090 | . . . 4 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥 ∨ 𝑥 = 𝑦)) | |
20 | df-3or 1084 | . . . 4 ⊢ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥 ∨ 𝑥 = 𝑦) ↔ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦)) | |
21 | 19, 20 | bitri 277 | . . 3 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦)) |
22 | 18, 21 | sylibr 236 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)) |
23 | 6, 22 | issod 5506 | 1 ⊢ (𝜑 → < Or 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∨ w3o 1082 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 ∪ cun 3934 ⊆ wss 3936 class class class wbr 5066 Po wpo 5472 Or wor 5473 × cxp 5553 ◡ccnv 5554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-po 5474 df-so 5475 df-xp 5561 df-cnv 5563 |
This theorem is referenced by: (None) |
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