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Mirrors > Home > MPE Home > Th. List > ordirr | Structured version Visualization version GIF 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 | ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordfr 5655 | . 2 ⊢ (Ord 𝐴 → E Fr 𝐴) | |
2 | efrirr 5019 | . 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 1977 E cep 4947 Fr wfr 4994 Ord word 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-eprel 4949 df-fr 4997 df-we 4999 df-ord 5643 |
This theorem is referenced by: nordeq 5659 ordn2lp 5660 ordtri3or 5672 ordtri1 5673 ordtri3 5676 ordtri3OLD 5677 orddisj 5679 ordunidif 5690 ordnbtwn 5733 ordnbtwnOLD 5734 onirri 5751 onssneli 5754 onprc 6876 nlimsucg 6934 nnlim 6970 limom 6972 smo11 7348 smoord 7349 tfrlem13 7373 omopth2 7551 limensuci 8021 infensuc 8023 ordtypelem9 8314 cantnfp1lem3 8460 cantnfp1 8461 oemapvali 8464 tskwe 8659 dif1card 8716 pm110.643ALT 8883 pwsdompw 8909 cflim2 8968 fin23lem24 9027 fin23lem26 9030 axdc3lem4 9158 ttukeylem7 9220 canthp1lem2 9354 inar1 9476 gruina 9519 grur1 9521 addnidpi 9602 fzennn 12629 hashp1i 13052 soseq 30995 noseponlem 31065 sucneqond 32389 |
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