Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4215 | . . 3 ⊢ ∪ 〈𝐴, 𝐵〉 = {𝐴, 𝐵} |
4 | 1, 2 | opi2 4193 | . . 3 ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
5 | 3, 4 | eqeltri 2230 | . 2 ⊢ ∪ 〈𝐴, 𝐵〉 ∈ 〈𝐴, 𝐵〉 |
6 | elssuni 3800 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 〈𝐴, 𝐵〉 ⊆ ∪ 𝐶) | |
7 | 6 | sseld 3127 | . 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 2128 Vcvv 2712 {cpr 3561 〈cop 3563 ∪ cuni 3772 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 |
This theorem is referenced by: dmrnssfld 4848 unielrel 5112 |
Copyright terms: Public domain | W3C validator |