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Mirrors > Home > MPE Home > Th. List > sorpssin | Structured version Visualization version GIF version |
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 790 | . . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴) | |
2 | df-ss 3554 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∩ 𝐶) = 𝐵) | |
3 | eleq1 2676 | . . . 4 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
4 | 2, 3 | sylbi 206 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) |
5 | 1, 4 | syl5ibrcom 236 | . 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 → (𝐵 ∩ 𝐶) ∈ 𝐴)) |
6 | simprr 792 | . . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ 𝐴) | |
7 | sseqin2 3779 | . . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶) | |
8 | eleq1 2676 | . . . 4 ⊢ ((𝐵 ∩ 𝐶) = 𝐶 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
9 | 7, 8 | sylbi 206 | . . 3 ⊢ (𝐶 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) |
10 | 6, 9 | syl5ibrcom 236 | . 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶 ⊆ 𝐵 → (𝐵 ∩ 𝐶) ∈ 𝐴)) |
11 | sorpssi 6841 | . 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) | |
12 | 5, 10, 11 | mpjaod 395 | 1 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∩ 𝐶) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 Or wor 4958 [⊊] crpss 6834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-so 4960 df-xp 5044 df-rel 5045 df-rpss 6835 |
This theorem is referenced by: (None) |
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