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Mirrors > Home > MPE Home > Th. List > xnegex | Structured version Visualization version Unicode version |
Description: A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xneg 11946 | . 2 | |
2 | mnfxr 10096 | . . . 4 | |
3 | 2 | elexi 3213 | . . 3 |
4 | pnfex 10093 | . . . 4 | |
5 | negex 10279 | . . . 4 | |
6 | 4, 5 | ifex 4156 | . . 3 |
7 | 3, 6 | ifex 4156 | . 2 |
8 | 1, 7 | eqeltri 2697 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 cvv 3200 cif 4086 cpnf 10071 cmnf 10072 cxr 10073 cneg 10267 cxne 11943 |
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-8 1992 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-pow 4843 ax-un 6949 ax-cnex 9992 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-uni 4437 df-iota 5851 df-fv 5896 df-ov 6653 df-pnf 10076 df-mnf 10077 df-xr 10078 df-neg 10269 df-xneg 11946 |
This theorem is referenced by: xrhmeo 22745 supminfxrrnmpt 39701 liminfvalxr 40015 |
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