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Mirrors > Home > MPE Home > Th. List > cnveq0 | Structured version Visualization version 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 5966 | . 2 ⊢ ◡∅ = ∅ | |
2 | rel0 5636 | . . . . 5 ⊢ Rel ∅ | |
3 | cnveqb 6020 | . . . . 5 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) | |
4 | 2, 3 | mpan2 690 | . . . 4 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) |
5 | eqeq2 2810 | . . . . 5 ⊢ (∅ = ◡∅ → (◡𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) | |
6 | 5 | bibi2d 346 | . . . 4 ⊢ (∅ = ◡∅ → ((𝐴 = ∅ ↔ ◡𝐴 = ∅) ↔ (𝐴 = ∅ ↔ ◡𝐴 = ◡∅))) |
7 | 4, 6 | syl5ibr 249 | . . 3 ⊢ (∅ = ◡∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))) |
8 | 7 | eqcoms 2806 | . 2 ⊢ (◡∅ = ∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))) |
9 | 1, 8 | ax-mp 5 | 1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∅c0 4243 ◡ccnv 5518 Rel wrel 5524 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 |
This theorem is referenced by: elrn3 33111 |
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