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Mirrors > Home > MPE Home > Th. List > rabn0 | Structured version Visualization version GIF version |
Description: Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999.) (Revised by BJ, 16-Jul-2021.) |
Ref | Expression |
---|---|
rabn0 | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeq0 4385 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
2 | 1 | necon3abii 2988 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) |
3 | dfrex2 3074 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
4 | 2, 3 | bitr4i 278 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3433 ∅c0 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-dif 3952 df-nul 4324 |
This theorem is referenced by: class2set 5354 reusv2 5402 exss 5464 frminex 5657 weniso 7351 onminesb 7781 onminsb 7782 onminex 7790 oeeulem 8601 supval2 9450 ordtypelem3 9515 card2on 9549 tz9.12lem3 9784 rankf 9789 scott0 9881 karden 9890 cardf2 9938 cardval3 9947 cardmin2 9994 acni3 10042 kmlem3 10147 cofsmo 10264 coftr 10268 fin23lem7 10311 enfin2i 10316 axcc4 10434 axdc3lem4 10448 ac6num 10474 pwfseqlem3 10655 wuncval 10737 wunccl 10739 tskmcl 10836 infm3 12173 nnwos 12899 zsupss 12921 zmin 12928 rpnnen1lem2 12961 rpnnen1lem1 12962 rpnnen1lem3 12963 rpnnen1lem5 12965 ioo0 13349 ico0 13370 ioc0 13371 icc0 13372 bitsfzolem 16375 lcmcllem 16533 fissn0dvdsn0 16557 odzcllem 16725 vdwnn 16931 ram0 16955 ramsey 16963 sylow2blem3 19490 iscyg2 19750 pgpfac1lem5 19949 ablfaclem2 19956 ablfaclem3 19957 ablfac 19958 lspf 20585 ordtrest2lem 22707 ordthauslem 22887 1stcfb 22949 2ndcdisj 22960 ptclsg 23119 txconn 23193 txflf 23510 tsmsfbas 23632 iscmet3 24810 minveclem3b 24945 iundisj 25065 dyadmax 25115 dyadmbllem 25116 elqaalem1 25832 elqaalem3 25834 sgmnncl 26651 musum 26695 conway 27301 incistruhgr 28370 uvtx01vtx 28685 spancl 30620 shsval2i 30671 ococin 30692 iundisjf 31851 iundisjfi 32038 ordtrest2NEWlem 32933 esumrnmpt2 33097 esumpinfval 33102 dmsigagen 33173 ballotlemfc0 33522 ballotlemfcc 33523 ballotlemiex 33531 ballotlemsup 33534 bnj110 33900 bnj1204 34054 bnj1253 34059 connpconn 34257 iscvm 34281 wsuclem 34828 poimirlem28 36564 sstotbnd2 36690 igenval 36977 igenidl 36979 pmap0 38684 aks4d1p4 40992 aks4d1p5 40993 aks4d1p7 40996 aks4d1p8 41000 fsuppind 41210 pellfundre 41667 pellfundge 41668 pellfundglb 41671 dgraalem 41935 rgspncl 41959 uzwo4 43788 ioodvbdlimc1lem1 44695 fourierdlem31 44902 fourierdlem64 44934 etransclem48 45046 subsaliuncl 45122 smflimlem6 45540 smfpimcc 45572 prmdvdsfmtnof1lem1 46300 prmdvdsfmtnof 46302 |
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