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| Description: A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.) | 
| Ref | Expression | 
|---|---|
| sorpssin | ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∩ 𝐶) ∈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simprl 771 | . . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴) | |
| 2 | dfss2 3969 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∩ 𝐶) = 𝐵) | |
| 3 | eleq1 2829 | . . . 4 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
| 4 | 2, 3 | sylbi 217 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | 
| 5 | 1, 4 | syl5ibrcom 247 | . 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 → (𝐵 ∩ 𝐶) ∈ 𝐴)) | 
| 6 | simprr 773 | . . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ 𝐴) | |
| 7 | sseqin2 4223 | . . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶) | |
| 8 | eleq1 2829 | . . . 4 ⊢ ((𝐵 ∩ 𝐶) = 𝐶 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
| 9 | 7, 8 | sylbi 217 | . . 3 ⊢ (𝐶 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | 
| 10 | 6, 9 | syl5ibrcom 247 | . 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶 ⊆ 𝐵 → (𝐵 ∩ 𝐶) ∈ 𝐴)) | 
| 11 | sorpssi 7749 | . 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) | |
| 12 | 5, 10, 11 | mpjaod 861 | 1 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∩ 𝐶) ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 Or wor 5591 [⊊] crpss 7742 | 
| 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-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-so 5593 df-xp 5691 df-rel 5692 df-rpss 7743 | 
| This theorem is referenced by: (None) | 
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