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Mirrors > Home > MPE Home > Th. List > ordirr | Structured version Visualization version Unicode version |
Description: Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. We prove this without invoking the Axiom of Regularity. (Contributed by NM, 2-Jan-1994.) |
Ref | Expression |
---|---|
ordirr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordfr 5738 | . 2 | |
2 | efrirr 5095 | . 2 | |
3 | 1, 2 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wcel 1990 cep 5028 wfr 5070 word 5722 |
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-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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-eprel 5029 df-fr 5073 df-we 5075 df-ord 5726 |
This theorem is referenced by: nordeq 5742 ordn2lp 5743 ordtri3or 5755 ordtri1 5756 ordtri3 5759 ordtri3OLD 5760 orddisj 5762 ordunidif 5773 ordnbtwn 5816 ordnbtwnOLD 5817 onirri 5834 onssneli 5837 onprc 6984 nlimsucg 7042 nnlim 7078 limom 7080 smo11 7461 smoord 7462 tfrlem13 7486 omopth2 7664 limensuci 8136 infensuc 8138 ordtypelem9 8431 cantnfp1lem3 8577 cantnfp1 8578 oemapvali 8581 tskwe 8776 dif1card 8833 pm110.643ALT 9000 pwsdompw 9026 cflim2 9085 fin23lem24 9144 fin23lem26 9147 axdc3lem4 9275 ttukeylem7 9337 canthp1lem2 9475 inar1 9597 gruina 9640 grur1 9642 addnidpi 9723 fzennn 12767 hashp1i 13191 soseq 31751 noseponlem 31817 noextend 31819 noextenddif 31821 noextendlt 31822 noextendgt 31823 fvnobday 31829 nosepssdm 31836 nosupbnd1lem3 31856 nosupbnd1lem5 31858 nosupbnd2lem1 31861 noetalem3 31865 sucneqond 33213 |
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