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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 4608 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
2 | elsni 4608 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴) | |
3 | 2 | eqcomd 2743 | . . . . . 6 ⊢ (𝑦 ∈ {𝐴} → 𝐴 = 𝑦) |
4 | 1, 3 | sylan9eq 2797 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦) |
5 | 4 | 3mix2d 1338 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) |
6 | 5 | rgen2 3195 | . . 3 ⊢ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) |
7 | df-so 5551 | . . 3 ⊢ (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
8 | 6, 7 | mpbiran2 709 | . 2 ⊢ (𝑅 Or {𝐴} ↔ 𝑅 Po {𝐴}) |
9 | posn 5722 | . 2 ⊢ (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴)) | |
10 | 8, 9 | bitrid 283 | 1 ⊢ (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ w3o 1087 ∈ wcel 2107 ∀wral 3065 {csn 4591 class class class wbr 5110 Po wpo 5548 Or wor 5549 Rel wrel 5643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 |
This theorem is referenced by: wesn 5725 |
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