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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 4587 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
2 | elsni 4587 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴) | |
3 | 2 | eqcomd 2830 | . . . . . 6 ⊢ (𝑦 ∈ {𝐴} → 𝐴 = 𝑦) |
4 | 1, 3 | sylan9eq 2879 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦) |
5 | 4 | 3mix2d 1333 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) |
6 | 5 | rgen2 3206 | . . 3 ⊢ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) |
7 | df-so 5478 | . . 3 ⊢ (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
8 | 6, 7 | mpbiran2 708 | . 2 ⊢ (𝑅 Or {𝐴} ↔ 𝑅 Po {𝐴}) |
9 | posn 5640 | . 2 ⊢ (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴)) | |
10 | 8, 9 | syl5bb 285 | 1 ⊢ (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ w3o 1082 ∈ wcel 2113 ∀wral 3141 {csn 4570 class class class wbr 5069 Po wpo 5475 Or wor 5476 Rel wrel 5563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 |
This theorem is referenced by: wesn 5643 |
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