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Mirrors > Home > MPE Home > Th. List > psstrd | Structured version Visualization version GIF version |
Description: Proper subclass inclusion is transitive. Deduction form of psstr 3995. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
psstrd.1 | ⊢ (𝜑 → 𝐴 ⊊ 𝐵) |
psstrd.2 | ⊢ (𝜑 → 𝐵 ⊊ 𝐶) |
Ref | Expression |
---|---|
psstrd | ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psstrd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊊ 𝐵) | |
2 | psstrd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊊ 𝐶) | |
3 | psstr 3995 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊊ wpss 3844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-v 3400 df-in 3850 df-ss 3860 df-pss 3862 |
This theorem is referenced by: (None) |
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