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Mirrors > Home > ILE Home > Th. List > cnveq0 | GIF version |
Description: A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.) |
Ref | Expression |
---|---|
cnveq0 | ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnv0 4747 | . 2 ⊢ ◡∅ = ∅ | |
2 | rel0 4480 | . . . . 5 ⊢ Rel ∅ | |
3 | cnveqb 4796 | . . . . 5 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) | |
4 | 2, 3 | mpan2 415 | . . . 4 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) |
5 | eqeq2 2090 | . . . . 5 ⊢ (∅ = ◡∅ → (◡𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) | |
6 | 5 | bibi2d 230 | . . . 4 ⊢ (∅ = ◡∅ → ((𝐴 = ∅ ↔ ◡𝐴 = ∅) ↔ (𝐴 = ∅ ↔ ◡𝐴 = ◡∅))) |
7 | 4, 6 | syl5ibr 154 | . . 3 ⊢ (∅ = ◡∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))) |
8 | 7 | eqcoms 2084 | . 2 ⊢ (◡∅ = ∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))) |
9 | 1, 8 | ax-mp 7 | 1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∅c0 3251 ◡ccnv 4362 Rel wrel 4368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 |
This theorem is referenced by: (None) |
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