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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 6066 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) | |
2 | dm0rn0 5797 | . . 3 ⊢ (dom {𝐴} = ∅ ↔ ran {𝐴} = ∅) | |
3 | 2 | necon3bii 3070 | . 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ran {𝐴} ≠ ∅) |
4 | 1, 3 | bitri 277 | 1 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∅c0 4293 {csn 4569 × cxp 5555 dom cdm 5557 ran crn 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 |
This theorem is referenced by: 2ndnpr 7696 2nd2val 7720 |
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