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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 3937. (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 3937 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | |
4 | 1, 2, 3 | syl2anc 581 | 1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊊ wpss 3799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-ne 3000 df-in 3805 df-ss 3812 df-pss 3814 |
This theorem is referenced by: (None) |
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