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Mirrors > Home > NFE Home > Th. List > sspss | GIF version |
Description: Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.) |
Ref | Expression |
---|---|
sspss | ⊢ (A ⊆ B ↔ (A ⊊ B ∨ A = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfpss2 3354 | . . . . 5 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ ¬ A = B)) | |
2 | 1 | simplbi2 608 | . . . 4 ⊢ (A ⊆ B → (¬ A = B → A ⊊ B)) |
3 | 2 | con1d 116 | . . 3 ⊢ (A ⊆ B → (¬ A ⊊ B → A = B)) |
4 | 3 | orrd 367 | . 2 ⊢ (A ⊆ B → (A ⊊ B ∨ A = B)) |
5 | pssss 3364 | . . 3 ⊢ (A ⊊ B → A ⊆ B) | |
6 | eqimss 3323 | . . 3 ⊢ (A = B → A ⊆ B) | |
7 | 5, 6 | jaoi 368 | . 2 ⊢ ((A ⊊ B ∨ A = B) → A ⊆ B) |
8 | 4, 7 | impbii 180 | 1 ⊢ (A ⊆ B ↔ (A ⊊ B ∨ A = B)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 176 ∨ wo 357 = wceq 1642 ⊆ 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: sspsstri 3371 sspsstr 3374 psssstr 3375 ssfin 4470 |
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