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Mirrors > Home > MPE Home > Th. List > uneqri | Structured version Visualization version GIF version |
Description: Inference from membership to union. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
uneqri.1 | ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) |
Ref | Expression |
---|---|
uneqri | ⊢ (𝐴 ∪ 𝐵) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 4063 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
2 | uneqri.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) | |
3 | 1, 2 | bitri 278 | . 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ 𝐶) |
4 | 3 | eqriv 2734 | 1 ⊢ (𝐴 ∪ 𝐵) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ∪ cun 3864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-un 3871 |
This theorem is referenced by: unidm 4066 uncom 4067 unass 4080 dfun2 4174 undi 4189 unab 4213 un0 4305 inundif 4393 |
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