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Mirrors > Home > MPE Home > Th. List > unidif | Structured version Visualization version GIF version |
Description: If the difference 𝐴 ∖ 𝐵 contains the largest members of 𝐴, then the union of the difference is the union of 𝐴. (Contributed by NM, 22-Mar-2004.) |
Ref | Expression |
---|---|
unidif | ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniss2 4889 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵)) | |
2 | difss 4078 | . . . 4 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
3 | 2 | unissi 4861 | . . 3 ⊢ ∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴 |
4 | 1, 3 | jctil 520 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → (∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵))) |
5 | eqss 3947 | . 2 ⊢ (∪ (𝐴 ∖ 𝐵) = ∪ 𝐴 ↔ (∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵))) | |
6 | 4, 5 | sylibr 233 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∀wral 3061 ∃wrex 3070 ∖ cdif 3895 ⊆ wss 3898 ∪ cuni 4852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3062 df-rex 3071 df-v 3443 df-dif 3901 df-in 3905 df-ss 3915 df-uni 4853 |
This theorem is referenced by: ordunidif 6350 |
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