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Mirrors > Home > MPE Home > Th. List > opnz | Structured version Visualization version Unicode version |
Description: An ordered pair is nonempty iff the arguments are sets. (Contributed by NM, 24-Jan-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opnz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprc 4424 | . . 3 | |
2 | 1 | necon1ai 2821 | . 2 |
3 | dfopg 4400 | . . 3 | |
4 | snex 4908 | . . . . 5 | |
5 | 4 | prnz 4310 | . . . 4 |
6 | 5 | a1i 11 | . . 3 |
7 | 3, 6 | eqnetrd 2861 | . 2 |
8 | 2, 7 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wcel 1990 wne 2794 cvv 3200 c0 3915 csn 4177 cpr 4179 cop 4183 |
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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 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 |
This theorem is referenced by: opnzi 4943 opeqex 4962 opelopabsb 4985 setsfun0 15894 |
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