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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 3581 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶) |
4 | 3 | ancoms 468 | 1 ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 |
This theorem is referenced by: intssuni2 4437 marypha1 8223 cardinfima 8803 cfflb 8964 ssfin4 9015 acsfn 16143 mrelatlub 17009 efgval 17953 islbs3 18976 kgentopon 21151 txlly 21249 sigaclci 29522 bnj1014 30284 topjoin 31530 filnetlem3 31545 poimirlem16 32595 mblfinlem3 32618 sspwimpALT 38183 sspwimpALT2 38186 |
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