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Mirrors > Home > ILE Home > Th. List > opeluu | GIF version |
Description: Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
opeluu.1 | ⊢ 𝐴 ∈ V |
opeluu.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opeluu | ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → (𝐴 ∈ ∪ ∪ 𝐶 ∧ 𝐵 ∈ ∪ ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeluu.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | prid1 3637 | . . 3 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
3 | opeluu.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
4 | 1, 3 | opi2 4163 | . . . 4 ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
5 | elunii 3749 | . . . 4 ⊢ (({𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 ∧ 〈𝐴, 𝐵〉 ∈ 𝐶) → {𝐴, 𝐵} ∈ ∪ 𝐶) | |
6 | 4, 5 | mpan 421 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → {𝐴, 𝐵} ∈ ∪ 𝐶) |
7 | elunii 3749 | . . 3 ⊢ ((𝐴 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐴 ∈ ∪ ∪ 𝐶) | |
8 | 2, 6, 7 | sylancr 411 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ ∪ ∪ 𝐶) |
9 | 3 | prid2 3638 | . . 3 ⊢ 𝐵 ∈ {𝐴, 𝐵} |
10 | elunii 3749 | . . 3 ⊢ ((𝐵 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐵 ∈ ∪ ∪ 𝐶) | |
11 | 9, 6, 10 | sylancr 411 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐵 ∈ ∪ ∪ 𝐶) |
12 | 8, 11 | jca 304 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → (𝐴 ∈ ∪ ∪ 𝐶 ∧ 𝐵 ∈ ∪ ∪ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1481 Vcvv 2689 {cpr 3533 〈cop 3535 ∪ cuni 3744 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 |
This theorem is referenced by: asymref 4932 |
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