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Mirrors > Home > MPE Home > Th. List > Mathboxes > socnv | Structured version Visualization version GIF version |
Description: The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.) |
Ref | Expression |
---|---|
socnv | ⊢ (𝑅 Or 𝐴 → ◡𝑅 Or 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvso 6310 | . 2 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
2 | 1 | biimpi 216 | 1 ⊢ (𝑅 Or 𝐴 → ◡𝑅 Or 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Or wor 5596 ◡ccnv 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-po 5597 df-so 5598 df-cnv 5697 |
This theorem is referenced by: (None) |
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