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Mirrors > Home > MPE Home > Th. List > sosn | Structured version Visualization version GIF version |
Description: Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
sosn | ⊢ (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsni 4544 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
2 | elsni 4544 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴) | |
3 | 2 | eqcomd 2742 | . . . . . 6 ⊢ (𝑦 ∈ {𝐴} → 𝐴 = 𝑦) |
4 | 1, 3 | sylan9eq 2791 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦) |
5 | 4 | 3mix2d 1339 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) |
6 | 5 | rgen2 3114 | . . 3 ⊢ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) |
7 | df-so 5454 | . . 3 ⊢ (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
8 | 6, 7 | mpbiran2 710 | . 2 ⊢ (𝑅 Or {𝐴} ↔ 𝑅 Po {𝐴}) |
9 | posn 5619 | . 2 ⊢ (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴)) | |
10 | 8, 9 | syl5bb 286 | 1 ⊢ (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ w3o 1088 ∈ wcel 2112 ∀wral 3051 {csn 4527 class class class wbr 5039 Po wpo 5451 Or wor 5452 Rel wrel 5541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 |
This theorem is referenced by: wesn 5622 |
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