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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 5535 | . 2 ⊢ ◡∅ = ∅ | |
2 | rel0 5243 | . . . . 5 ⊢ Rel ∅ | |
3 | cnveqb 5590 | . . . . 5 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) | |
4 | 2, 3 | mpan2 707 | . . . 4 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) |
5 | eqeq2 2633 | . . . . 5 ⊢ (∅ = ◡∅ → (◡𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) | |
6 | 5 | bibi2d 332 | . . . 4 ⊢ (∅ = ◡∅ → ((𝐴 = ∅ ↔ ◡𝐴 = ∅) ↔ (𝐴 = ∅ ↔ ◡𝐴 = ◡∅))) |
7 | 4, 6 | syl5ibr 236 | . . 3 ⊢ (∅ = ◡∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))) |
8 | 7 | eqcoms 2630 | . 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 196 = wceq 1483 ∅c0 3915 ◡ccnv 5113 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 |
This theorem is referenced by: elrn3 31652 |
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