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| Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.) | 
| Ref | Expression | 
|---|---|
| sspsstr | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sspss 4102 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵)) | |
| 2 | psstr 4107 | . . . . 5 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | |
| 3 | 2 | ex 412 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶)) | 
| 4 | psseq1 4090 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) | |
| 5 | 4 | biimprd 248 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶)) | 
| 6 | 3, 5 | jaoi 858 | . . 3 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶)) | 
| 7 | 6 | imp 406 | . 2 ⊢ (((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | 
| 8 | 1, 7 | sylanb 581 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ⊆ 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: sspsstrd 4111 ordtr2 6428 php 9247 phpOLD 9259 canthp1lem2 10693 suplem1pr 11092 fbfinnfr 23849 ppiltx 27220 | 
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