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Mirrors > Home > MPE Home > Th. List > uniopel | Structured version Visualization version GIF version |
Description: Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opthw.1 | ⊢ 𝐴 ∈ V |
opthw.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
uniopel | ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → ∪ 〈𝐴, 𝐵〉 ∈ ∪ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opthw.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | opthw.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | uniop 4902 | . . 3 ⊢ ∪ 〈𝐴, 𝐵〉 = {𝐴, 𝐵} |
4 | 1, 2 | opi2 4865 | . . 3 ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
5 | 3, 4 | eqeltri 2684 | . 2 ⊢ ∪ 〈𝐴, 𝐵〉 ∈ 〈𝐴, 𝐵〉 |
6 | elssuni 4403 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 〈𝐴, 𝐵〉 ⊆ ∪ 𝐶) | |
7 | 6 | sseld 3567 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → (∪ 〈𝐴, 𝐵〉 ∈ 〈𝐴, 𝐵〉 → ∪ 〈𝐴, 𝐵〉 ∈ ∪ 𝐶)) |
8 | 5, 7 | mpi 20 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → ∪ 〈𝐴, 𝐵〉 ∈ ∪ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 {cpr 4127 〈cop 4131 ∪ cuni 4372 |
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-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 |
This theorem is referenced by: dmrnssfld 5305 unielrel 5577 |
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