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Mirrors > Home > MPE Home > Th. List > iunconst | Structured version Visualization version GIF version |
Description: Indexed union of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
iunconst | ⊢ (𝐴 ≠ ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliun 5005 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | r19.9rzv 4504 | . . 3 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
3 | 1, 2 | bitr4id 289 | . 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ 𝐵)) |
4 | 3 | eqrdv 2724 | 1 ⊢ (𝐴 ≠ ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 ∅c0 4325 ∪ ciun 5001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-v 3464 df-dif 3950 df-nul 4326 df-iun 5003 |
This theorem is referenced by: iununi 5107 oe1m 8575 oarec 8592 oelim2 8625 bnj1143 34635 poimirlem32 37353 mblfinlem2 37359 scottrankd 43922 hoicvr 46169 ovnlecvr2 46231 iunhoiioo 46297 |
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