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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 3977 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶) |
4 | 3 | ancoms 459 | 1 ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-in 3940 df-ss 3949 |
This theorem is referenced by: intssuni2 4892 marypha1 8886 cardinfima 9511 cfflb 9669 ssfin4 9720 acsfn 16918 mrelatlub 17784 efgval 18772 islbs3 19856 kgentopon 22074 txlly 22172 sigaclci 31290 bnj1014 32131 topjoin 33610 filnetlem3 33625 poimirlem16 34789 mblfinlem3 34812 sspwimpALT 41136 sspwimpALT2 41139 setrecsres 44732 |
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