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Mirrors > Home > NFE Home > Th. List > ssnpss | GIF version |
Description: Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
ssnpss | ⊢ (A ⊆ B → ¬ B ⊊ A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfpss3 3355 | . . 3 ⊢ (B ⊊ A ↔ (B ⊆ A ∧ ¬ A ⊆ B)) | |
2 | 1 | simprbi 450 | . 2 ⊢ (B ⊊ A → ¬ A ⊆ B) |
3 | 2 | con2i 112 | 1 ⊢ (A ⊆ B → ¬ B ⊊ A) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ⊆ wss 3257 ⊊ wpss 3258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-pss 3261 |
This theorem is referenced by: (None) |
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