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Mirrors > Home > MPE Home > Th. List > sylan9ssr | Structured version Visualization version GIF version |
Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) |
Ref | Expression |
---|---|
sylan9ssr.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sylan9ssr.2 | ⊢ (𝜓 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
sylan9ssr | ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylan9ssr.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sylan9ssr.2 | . . 3 ⊢ (𝜓 → 𝐵 ⊆ 𝐶) | |
3 | 1, 2 | sylan9ss 3980 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶) |
4 | 3 | ancoms 461 | 1 ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ⊆ wss 3936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3943 df-ss 3952 |
This theorem is referenced by: intssuni2 4901 marypha1 8898 cardinfima 9523 cfflb 9681 ssfin4 9732 acsfn 16930 mrelatlub 17796 efgval 18843 islbs3 19927 kgentopon 22146 txlly 22244 sigaclci 31391 bnj1014 32233 topjoin 33713 filnetlem3 33728 poimirlem16 34923 mblfinlem3 34946 sspwimpALT 41279 sspwimpALT2 41282 setrecsres 44824 |
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