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Mirrors > Home > ILE Home > Th. List > ssnel | GIF version |
Description: Relationship between subset and elementhood. In the context of ordinals this can be seen as an ordering law. (Contributed by Jim Kingdon, 22-Jul-2019.) |
Ref | Expression |
---|---|
ssnel | ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirr 4319 | . 2 ⊢ ¬ 𝐵 ∈ 𝐵 | |
2 | ssel 3004 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐵)) | |
3 | 1, 2 | mtoi 623 | 1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1434 ⊆ wss 2984 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-setind 4315 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-v 2614 df-dif 2986 df-in 2990 df-ss 2997 df-sn 3428 |
This theorem is referenced by: nntri1 6188 |
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