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Mirrors > Home > MPE Home > Th. List > opeluu | Structured version Visualization version 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 4658 | . . 3 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
3 | opeluu.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
4 | 1, 3 | opi2 5326 | . . . 4 ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
5 | elunii 4805 | . . . 4 ⊢ (({𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 ∧ 〈𝐴, 𝐵〉 ∈ 𝐶) → {𝐴, 𝐵} ∈ ∪ 𝐶) | |
6 | 4, 5 | mpan 689 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → {𝐴, 𝐵} ∈ ∪ 𝐶) |
7 | elunii 4805 | . . 3 ⊢ ((𝐴 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐴 ∈ ∪ ∪ 𝐶) | |
8 | 2, 6, 7 | sylancr 590 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ ∪ ∪ 𝐶) |
9 | 3 | prid2 4659 | . . 3 ⊢ 𝐵 ∈ {𝐴, 𝐵} |
10 | elunii 4805 | . . 3 ⊢ ((𝐵 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐵 ∈ ∪ ∪ 𝐶) | |
11 | 9, 6, 10 | sylancr 590 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐵 ∈ ∪ ∪ 𝐶) |
12 | 8, 11 | jca 515 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → (𝐴 ∈ ∪ ∪ 𝐶 ∧ 𝐵 ∈ ∪ ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 {cpr 4527 〈cop 4531 ∪ cuni 4800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 |
This theorem is referenced by: asymref 5943 asymref2 5944 wrdexb 13868 |
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