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Mirrors > Home > ILE Home > Th. List > nssne2 | GIF version |
Description: Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.) |
Ref | Expression |
---|---|
nssne2 | ⊢ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐵 ⊆ 𝐶) → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3193 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
2 | 1 | biimpcd 159 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 = 𝐵 → 𝐵 ⊆ 𝐶)) |
3 | 2 | necon3bd 2403 | . 2 ⊢ (𝐴 ⊆ 𝐶 → (¬ 𝐵 ⊆ 𝐶 → 𝐴 ≠ 𝐵)) |
4 | 3 | imp 124 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐵 ⊆ 𝐶) → 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 = wceq 1364 ≠ wne 2360 ⊆ wss 3144 |
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 615 ax-in2 616 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-ne 2361 df-in 3150 df-ss 3157 |
This theorem is referenced by: (None) |
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