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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 4297 | . . 3 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
3 | opeluu.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
4 | 1, 3 | opi2 4938 | . . . 4 ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
5 | elunii 4441 | . . . 4 ⊢ (({𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 ∧ 〈𝐴, 𝐵〉 ∈ 𝐶) → {𝐴, 𝐵} ∈ ∪ 𝐶) | |
6 | 4, 5 | mpan 706 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → {𝐴, 𝐵} ∈ ∪ 𝐶) |
7 | elunii 4441 | . . 3 ⊢ ((𝐴 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐴 ∈ ∪ ∪ 𝐶) | |
8 | 2, 6, 7 | sylancr 695 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ ∪ ∪ 𝐶) |
9 | 3 | prid2 4298 | . . 3 ⊢ 𝐵 ∈ {𝐴, 𝐵} |
10 | elunii 4441 | . . 3 ⊢ ((𝐵 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐵 ∈ ∪ ∪ 𝐶) | |
11 | 9, 6, 10 | sylancr 695 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐵 ∈ ∪ ∪ 𝐶) |
12 | 8, 11 | jca 554 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → (𝐴 ∈ ∪ ∪ 𝐶 ∧ 𝐵 ∈ ∪ ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 Vcvv 3200 {cpr 4179 〈cop 4183 ∪ cuni 4436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 |
This theorem is referenced by: asymref 5512 asymref2 5513 wrdexb 13316 |
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