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| Description: Proper subclass inclusion is transitive. Deduction form of psstr 4107. (Contributed by David Moews, 1-May-2017.) | 
| Ref | Expression | 
|---|---|
| psstrd.1 | ⊢ (𝜑 → 𝐴 ⊊ 𝐵) | 
| psstrd.2 | ⊢ (𝜑 → 𝐵 ⊊ 𝐶) | 
| Ref | Expression | 
|---|---|
| psstrd | ⊢ (𝜑 → 𝐴 ⊊ 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | psstrd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊊ 𝐵) | |
| 2 | psstrd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊊ 𝐶) | |
| 3 | psstr 4107 | . 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 ⊊ 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-ex 1780 df-cleq 2729 df-ne 2941 df-ss 3968 df-pss 3971 | 
| This theorem is referenced by: (None) | 
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