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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 4080. (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 4080 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊊ wpss 3936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-ne 3017 df-in 3942 df-ss 3951 df-pss 3953 |
This theorem is referenced by: (None) |
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