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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 5396 | . . 3 ⊢ ∪ 〈𝐴, 𝐵〉 = {𝐴, 𝐵} |
4 | 1, 2 | opi2 5352 | . . 3 ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
5 | 3, 4 | eqeltri 2906 | . 2 ⊢ ∪ 〈𝐴, 𝐵〉 ∈ 〈𝐴, 𝐵〉 |
6 | elssuni 4859 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 〈𝐴, 𝐵〉 ⊆ ∪ 𝐶) | |
7 | 6 | sseld 3963 | . 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 2105 Vcvv 3492 {cpr 4559 〈cop 4563 ∪ cuni 4830 |
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-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 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 |
This theorem is referenced by: dmrnssfld 5834 unielrel 6118 |
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