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Mirrors > Home > MPE Home > Th. List > sspsstrd | Structured version Visualization version GIF version |
Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3712. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
sspsstrd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sspsstrd.2 | ⊢ (𝜑 → 𝐵 ⊊ 𝐶) |
Ref | Expression |
---|---|
sspsstrd | ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspsstrd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sspsstrd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊊ 𝐶) | |
3 | sspsstr 3712 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | |
4 | 1, 2, 3 | syl2anc 693 | 1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3574 ⊊ wpss 3575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-ne 2795 df-in 3581 df-ss 3588 df-pss 3590 |
This theorem is referenced by: marypha1lem 8339 ackbij1lem15 9056 fin23lem38 9171 ltexprlem2 9859 mrieqv2d 16299 |
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