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Mirrors > Home > NFE Home > Th. List > sstr2 | GIF version |
Description: Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Ref | Expression |
---|---|
sstr2 | ⊢ (A ⊆ B → (B ⊆ C → A ⊆ C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3267 | . . . 4 ⊢ (A ⊆ B → (x ∈ A → x ∈ B)) | |
2 | 1 | imim1d 69 | . . 3 ⊢ (A ⊆ B → ((x ∈ B → x ∈ C) → (x ∈ A → x ∈ C))) |
3 | 2 | alimdv 1621 | . 2 ⊢ (A ⊆ B → (∀x(x ∈ B → x ∈ C) → ∀x(x ∈ A → x ∈ C))) |
4 | dfss2 3262 | . 2 ⊢ (B ⊆ C ↔ ∀x(x ∈ B → x ∈ C)) | |
5 | dfss2 3262 | . 2 ⊢ (A ⊆ C ↔ ∀x(x ∈ A → x ∈ C)) | |
6 | 3, 4, 5 | 3imtr4g 261 | 1 ⊢ (A ⊆ B → (B ⊆ C → A ⊆ C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∈ wcel 1710 ⊆ wss 3257 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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: sstr 3280 sstri 3281 sseq1 3292 sseq2 3293 ssun3 3428 ssun4 3429 ssinss1 3483 ssdisj 3600 sspwb 4118 funss 5126 funimass2 5170 fss 5230 |
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