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Mirrors > Home > MPE Home > Th. List > sspsstrd | Structured version Visualization version GIF version |
Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4105. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
sspsstrd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sspsstrd.2 | ⊢ (𝜑 → 𝐵 ⊊ 𝐶) |
Ref | Expression |
---|---|
sspsstrd | ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspsstrd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sspsstrd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊊ 𝐶) | |
3 | sspsstr 4105 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3948 ⊊ wpss 3949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-v 3476 df-in 3955 df-ss 3965 df-pss 3967 |
This theorem is referenced by: marypha1lem 9427 ackbij1lem15 10228 fin23lem38 10343 ltexprlem2 11031 mrieqv2d 17582 |
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