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Mirrors > Home > HSE Home > Th. List > dmdi2 | Structured version Visualization version GIF version |
Description: Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dmdi2 | ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmdi 30006 | . 2 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵))) | |
2 | eqimss2 4021 | . 2 ⊢ (((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 class class class wbr 5057 (class class class)co 7145 Cℋ cch 28633 ∨ℋ chj 28637 𝑀ℋ* cdmd 28671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-iota 6307 df-fv 6356 df-ov 7148 df-dmd 29985 |
This theorem is referenced by: (None) |
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