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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 2808, ax-8 2106. (Revised by Gino Giotto, 2-Sep-2024.) |
Ref | Expression |
---|---|
ral0 | ⊢ ∀𝑥 ∈ ∅ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2730 | . 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 1539 ∀wral 3059 ∅c0 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-ral 3060 df-dif 3952 df-nul 4324 |
This theorem is referenced by: int0 4967 0iin 5068 po0 5606 so0 5625 mpt0 6693 naddrid 8686 ixp0x 8924 ac6sfi 9291 sup0riota 9464 infpssrlem4 10305 axdc3lem4 10452 0tsk 10754 uzsupss 12930 xrsupsslem 13292 xrinfmsslem 13293 xrsup0 13308 fsuppmapnn0fiubex 13963 swrd0 14614 swrdspsleq 14621 repswsymballbi 14736 cshw1 14778 rexfiuz 15300 lcmf0 16577 2prm 16635 0ssc 17793 0subcat 17794 drsdirfi 18264 0pos 18280 0posOLD 18281 mrelatglb0 18520 sgrp0b 18655 ga0 19205 psgnunilem3 19407 lbsexg 20924 ocv0 21451 mdetunilem9 22344 imasdsf1olem 24101 prdsxmslem2 24260 lebnumlem3 24711 cniccbdd 25212 ovolicc2lem4 25271 c1lip1 25748 ulm0 26137 precsexlem9 27898 istrkg2ld 27976 nbgr0vtx 28878 nbgr1vtx 28880 cplgr0 28947 cplgr1v 28952 wwlksn0s 29380 clwwlkn 29544 clwwlkn1 29559 0ewlk 29632 1ewlk 29633 0wlk 29634 0conngr 29710 frgr0v 29780 frgr0 29783 frgr1v 29789 1vwmgr 29794 chocnul 30846 locfinref 33117 esumnul 33342 derang0 34456 unt0 34982 fdc 36918 lub0N 38364 glb0N 38368 0psubN 38925 sticksstones11 41280 cantnfresb 42378 safesnsupfilb 42473 nla0002 42479 nla0003 42480 iso0 43370 fnchoice 44017 eliuniincex 44101 eliincex 44102 limcdm0 44634 2ffzoeq 46336 iccpartiltu 46390 iccpartigtl 46391 0mgm 46844 linds0 47235 0thincg 47759 |
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