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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 6238 | . 2 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
2 | 1 | biimpi 215 | 1 ⊢ (𝑅 Or 𝐴 → ◡𝑅 Or 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Or wor 5542 ◡ccnv 5630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-po 5543 df-so 5544 df-cnv 5639 |
This theorem is referenced by: (None) |
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