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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 6228 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) | |
2 | dm0rn0 5937 | . . 3 ⊢ (dom {𝐴} = ∅ ↔ ran {𝐴} = ∅) | |
3 | 2 | necon3bii 2990 | . 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 2105 ≠ wne 2937 Vcvv 3477 ∅c0 4338 {csn 4630 × cxp 5686 dom cdm 5688 ran crn 5689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-dm 5698 df-rn 5699 |
This theorem is referenced by: 2ndnpr 8017 2nd2val 8041 |
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