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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. Theorem 1.9(i) of [Schloeder] p. 1. 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 6399 | . 2 ⊢ (Ord 𝐴 → E Fr 𝐴) | |
| 2 | efrirr 5665 | . 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 2108 E cep 5583 Fr wfr 5634 Ord word 6383 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-fr 5637 df-we 5639 df-ord 6387 |
| This theorem is referenced by: nordeq 6403 ordn2lp 6404 ordtri3or 6416 ordtri1 6417 ordtri3 6420 orddisj 6422 ordunidif 6433 ordnbtwn 6477 onirri 6497 onssneli 6500 epweon 7795 onprc 7798 nlimsucg 7863 nnlim 7901 limom 7903 soseq 8184 smo11 8404 smoord 8405 tfrlem13 8430 omopth2 8622 cofonr 8712 naddcllem 8714 limensuci 9193 infensuc 9195 ordtypelem9 9566 cantnfp1lem3 9720 cantnfp1 9721 oemapvali 9724 tskwe 9990 dif1card 10050 dju1p1e2ALT 10215 nnadju 10238 pwsdompw 10243 cflim2 10303 fin23lem24 10362 fin23lem26 10365 axdc3lem4 10493 ttukeylem7 10555 canthp1lem2 10693 inar1 10815 gruina 10858 grur1 10860 addnidpi 10941 fzennn 14009 hashp1i 14442 noseponlem 27709 noextend 27711 noextenddif 27713 noextendlt 27714 noextendgt 27715 fvnobday 27723 nosepssdm 27731 nosupbnd1lem3 27755 nosupbnd1lem5 27757 nosupbnd2lem1 27760 noinfbnd1lem3 27770 noinfbnd1lem5 27772 noinfbnd2lem1 27775 noetasuplem4 27781 noetainflem4 27785 sucneqond 37366 oaordnrex 43308 omnord1ex 43317 oenord1ex 43328 cantnfresb 43337 tfsconcatb0 43357 nlimsuc 43454 |
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