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Mirrors > Home > ILE Home > Th. List > uniopel | 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 4240 | . . 3 ⊢ ∪ 〈𝐴, 𝐵〉 = {𝐴, 𝐵} |
4 | 1, 2 | opi2 4218 | . . 3 ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
5 | 3, 4 | eqeltri 2243 | . 2 ⊢ ∪ 〈𝐴, 𝐵〉 ∈ 〈𝐴, 𝐵〉 |
6 | elssuni 3824 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 〈𝐴, 𝐵〉 ⊆ ∪ 𝐶) | |
7 | 6 | sseld 3146 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → (∪ 〈𝐴, 𝐵〉 ∈ 〈𝐴, 𝐵〉 → ∪ 〈𝐴, 𝐵〉 ∈ ∪ 𝐶)) |
8 | 5, 7 | mpi 15 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → ∪ 〈𝐴, 𝐵〉 ∈ ∪ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 Vcvv 2730 {cpr 3584 〈cop 3586 ∪ cuni 3796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 |
This theorem is referenced by: dmrnssfld 4874 unielrel 5138 |
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