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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 7561 | . 2 ⊢ 2𝑜 = suc 1𝑜 | |
2 | 1on 7567 | . . 3 ⊢ 1𝑜 ∈ On | |
3 | 2 | onsuci 7038 | . 2 ⊢ suc 1𝑜 ∈ On |
4 | 1, 3 | eqeltri 2697 | 1 ⊢ 2𝑜 ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 Oncon0 5723 suc csuc 5725 1𝑜c1o 7553 2𝑜c2o 7554 |
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-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-suc 5729 df-1o 7560 df-2o 7561 |
This theorem is referenced by: 3on 7570 o2p2e4 7621 oneo 7661 infxpenc 8841 infxpenc2 8845 mappwen 8935 pwcdaen 9007 sdom2en01 9124 fin1a2lem4 9225 xpslem 16233 xpsadd 16236 xpsmul 16237 xpsvsca 16239 xpsle 16241 xpsmnd 17330 xpsgrp 17534 efgval 18130 efgtf 18135 frgpcpbl 18172 frgp0 18173 frgpeccl 18174 frgpadd 18176 frgpmhm 18178 vrgpf 18181 vrgpinv 18182 frgpupf 18186 frgpup1 18188 frgpup2 18189 frgpup3lem 18190 frgpnabllem1 18276 frgpnabllem2 18277 xpstopnlem1 21612 xpstps 21613 xpstopnlem2 21614 xpsxmetlem 22184 xpsdsval 22186 nofv 31810 sltres 31815 noextendgt 31823 nolesgn2ores 31825 nosepnelem 31830 nosepdmlem 31833 nolt02o 31845 nosupno 31849 nosupbday 31851 nosupbnd1lem3 31856 nosupbnd1 31860 nosupbnd2lem1 31861 nosupbnd2 31862 ssoninhaus 32447 onint1 32448 1oequni2o 33216 finxpreclem4 33231 pw2f1ocnv 37604 frlmpwfi 37668 enrelmap 38291 clsk1indlem1 38343 clsk1independent 38344 |
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