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Mirrors > Home > MPE Home > Th. List > ordirr | Structured version Visualization version GIF version |
Description: No ordinal class is a member of itself. In other words, the membership relation is irreflexive on ordinal classes. 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 | ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordfr 6228 | . 2 ⊢ (Ord 𝐴 → E Fr 𝐴) | |
2 | efrirr 5532 | . 2 ⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2110 E cep 5459 Fr wfr 5506 Ord word 6212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-eprel 5460 df-fr 5509 df-we 5511 df-ord 6216 |
This theorem is referenced by: nordeq 6232 ordn2lp 6233 ordtri3or 6245 ordtri1 6246 ordtri3 6249 orddisj 6251 ordunidif 6261 ordnbtwn 6303 onirri 6320 onssneli 6323 onprc 7562 nlimsucg 7621 nnlim 7658 limom 7660 smo11 8101 smoord 8102 tfrlem13 8126 omopth2 8312 limensuci 8822 infensuc 8824 ordtypelem9 9142 cantnfp1lem3 9295 cantnfp1 9296 oemapvali 9299 tskwe 9566 dif1card 9624 dju1p1e2ALT 9788 nnadju 9811 pwsdompw 9818 cflim2 9877 fin23lem24 9936 fin23lem26 9939 axdc3lem4 10067 ttukeylem7 10129 canthp1lem2 10267 inar1 10389 gruina 10432 grur1 10434 addnidpi 10515 fzennn 13541 hashp1i 13970 soseq 33540 naddcllem 33568 noseponlem 33604 noextend 33606 noextenddif 33608 noextendlt 33609 noextendgt 33610 fvnobday 33618 nosepssdm 33626 nosupbnd1lem3 33650 nosupbnd1lem5 33652 nosupbnd2lem1 33655 noinfbnd1lem3 33665 noinfbnd1lem5 33667 noinfbnd2lem1 33670 noetasuplem4 33676 noetainflem4 33680 sucneqond 35273 |
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