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| Mirrors > Home > MPE Home > Th. List > rnsnn0 | Structured version Visualization version GIF version | ||
| Description: The range of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) |
| Ref | Expression |
|---|---|
| rnsnn0 | ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmsnn0 6227 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) | |
| 2 | dm0rn0 5935 | . . 3 ⊢ (dom {𝐴} = ∅ ↔ ran {𝐴} = ∅) | |
| 3 | 2 | necon3bii 2993 | . 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ran {𝐴} ≠ ∅) |
| 4 | 1, 3 | bitri 275 | 1 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 {csn 4626 × cxp 5683 dom cdm 5685 ran crn 5686 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: 2ndnpr 8019 2nd2val 8043 |
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