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Mirrors > Home > MPE Home > Th. List > ral0 | Structured version Visualization version GIF version |
Description: Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) Avoid df-clel 2811, ax-8 2109. (Revised by Gino Giotto, 2-Sep-2024.) |
Ref | Expression |
---|---|
ral0 | ⊢ ∀𝑥 ∈ ∅ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . 2 ⊢ ∅ = ∅ | |
2 | rzal 4509 | . 2 ⊢ (∅ = ∅ → ∀𝑥 ∈ ∅ 𝜑) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∀𝑥 ∈ ∅ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∀wral 3062 ∅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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-ral 3063 df-dif 3952 df-nul 4324 |
This theorem is referenced by: int0 4967 0iin 5068 po0 5606 so0 5625 mpt0 6693 naddrid 8682 ixp0x 8920 ac6sfi 9287 sup0riota 9460 infpssrlem4 10301 axdc3lem4 10448 0tsk 10750 uzsupss 12924 xrsupsslem 13286 xrinfmsslem 13287 xrsup0 13302 fsuppmapnn0fiubex 13957 swrd0 14608 swrdspsleq 14615 repswsymballbi 14730 cshw1 14772 rexfiuz 15294 lcmf0 16571 2prm 16629 0ssc 17787 0subcat 17788 drsdirfi 18258 0pos 18274 0posOLD 18275 mrelatglb0 18514 sgrp0b 18619 ga0 19162 psgnunilem3 19364 lbsexg 20777 ocv0 21230 mdetunilem9 22122 imasdsf1olem 23879 prdsxmslem2 24038 lebnumlem3 24479 cniccbdd 24978 ovolicc2lem4 25037 c1lip1 25514 ulm0 25903 precsexlem9 27661 istrkg2ld 27711 nbgr0vtx 28613 nbgr1vtx 28615 cplgr0 28682 cplgr1v 28687 wwlksn0s 29115 clwwlkn 29279 clwwlkn1 29294 0ewlk 29367 1ewlk 29368 0wlk 29369 0conngr 29445 frgr0v 29515 frgr0 29518 frgr1v 29524 1vwmgr 29529 chocnul 30581 locfinref 32821 esumnul 33046 derang0 34160 unt0 34680 fdc 36613 lub0N 38059 glb0N 38063 0psubN 38620 sticksstones11 40972 cantnfresb 42074 safesnsupfilb 42169 nla0002 42175 nla0003 42176 iso0 43066 fnchoice 43713 eliuniincex 43798 eliincex 43799 limcdm0 44334 2ffzoeq 46036 iccpartiltu 46090 iccpartigtl 46091 0mgm 46544 linds0 47146 0thincg 47670 |
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