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Mirrors > Home > ILE Home > Th. List > onirri | GIF version |
Description: An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
Ref | Expression |
---|---|
onirri.1 | ⊢ 𝐴 ∈ On |
Ref | Expression |
---|---|
onirri | ⊢ ¬ 𝐴 ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onirri.1 | . . 3 ⊢ 𝐴 ∈ On | |
2 | 1 | onordi 4428 | . 2 ⊢ Ord 𝐴 |
3 | ordirr 4543 | . 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ¬ 𝐴 ∈ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2148 Ord word 4364 Oncon0 4365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-setind 4538 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2741 df-dif 3133 df-in 3137 df-ss 3144 df-sn 3600 df-uni 3812 df-tr 4104 df-iord 4368 df-on 4370 |
This theorem is referenced by: ontri2orexmidim 4573 enpr2d 6819 pm54.43 7191 pw1ne1 7230 |
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