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Mirrors > Home > NFE Home > Th. List > syl5sseq | GIF version |
Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
syl5sseq.1 | ⊢ B ⊆ A |
syl5sseq.2 | ⊢ (φ → A = C) |
Ref | Expression |
---|---|
syl5sseq | ⊢ (φ → B ⊆ C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl5sseq.2 | . 2 ⊢ (φ → A = C) | |
2 | syl5sseq.1 | . 2 ⊢ B ⊆ A | |
3 | sseq2 3293 | . . 3 ⊢ (A = C → (B ⊆ A ↔ B ⊆ C)) | |
4 | 3 | biimpa 470 | . 2 ⊢ ((A = C ∧ B ⊆ A) → B ⊆ C) |
5 | 1, 2, 4 | sylancl 643 | 1 ⊢ (φ → B ⊆ C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ⊆ wss 3257 |
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-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 |
This theorem is referenced by: fconst4 5458 ecss 5966 sbthlem3 6205 |
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