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| Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4108. (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 4108 | . 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 3951 ⊊ wpss 3952 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-cleq 2729 df-ne 2941 df-ss 3968 df-pss 3971 | 
| This theorem is referenced by: marypha1lem 9473 ackbij1lem15 10273 fin23lem38 10389 ltexprlem2 11077 mrieqv2d 17682 | 
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