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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 4385 | . 2 ⊢ ¬ 𝐵 ∈ 𝐵 | |
2 | ssel 3033 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐵)) | |
3 | 1, 2 | mtoi 628 | 1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1445 ⊆ wss 3013 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-setind 4381 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-v 2635 df-dif 3015 df-in 3019 df-ss 3026 df-sn 3472 |
This theorem is referenced by: nntri1 6297 |
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