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Mirrors > Home > MPE Home > Th. List > 2on | Structured version Visualization version GIF version |
Description: Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
2on | ⊢ 2o ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2o 8094 | . 2 ⊢ 2o = suc 1o | |
2 | 1on 8100 | . . 3 ⊢ 1o ∈ On | |
3 | 2 | onsuci 7541 | . 2 ⊢ suc 1o ∈ On |
4 | 1, 3 | eqeltri 2909 | 1 ⊢ 2o ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 Oncon0 6185 suc csuc 6187 1oc1o 8086 2oc2o 8087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-tr 5165 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-ord 6188 df-on 6189 df-suc 6191 df-1o 8093 df-2o 8094 |
This theorem is referenced by: 3on 8105 o2p2e4 8157 oneo 8197 nneob 8269 infxpenc 9433 infxpenc2 9437 mappwen 9527 pwdjuen 9596 ackbij1lem5 9635 sdom2en01 9713 fin1a2lem4 9814 fin1a2lem6 9816 xpsrnbas 16834 xpsadd 16837 xpsmul 16838 xpsvsca 16840 xpsle 16842 xpsmnd 17941 xpsgrp 18158 efgval 18774 efgtf 18779 frgpcpbl 18816 frgp0 18817 frgpeccl 18818 frgpadd 18820 frgpmhm 18822 vrgpf 18825 vrgpinv 18826 frgpupf 18830 frgpup1 18832 frgpup2 18833 frgpup3lem 18834 frgpnabllem1 18924 frgpnabllem2 18925 xpstopnlem1 22347 xpstps 22348 xpstopnlem2 22349 xpsxmetlem 22918 xpsdsval 22920 nofv 33062 sltres 33067 noextendgt 33075 nolesgn2ores 33077 nosepnelem 33082 nosepdmlem 33085 nolt02o 33097 nosupno 33101 nosupbday 33103 nosupbnd1lem3 33108 nosupbnd1 33112 nosupbnd2lem1 33113 nosupbnd2 33114 ssoninhaus 33694 onint1 33695 1oequni2o 34532 finxpreclem4 34558 pw2f1ocnv 39514 frlmpwfi 39578 tr3dom 39774 enrelmap 40223 |
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