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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 4118. (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 4118 | . 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 3963 ⊊ wpss 3964 |
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 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-cleq 2727 df-ne 2939 df-ss 3980 df-pss 3983 |
This theorem is referenced by: marypha1lem 9471 ackbij1lem15 10271 fin23lem38 10387 ltexprlem2 11075 mrieqv2d 17684 |
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