![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4767 | . . 3 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
3 | opeluu.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
4 | 1, 3 | opi2 5470 | . . . 4 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩ |
5 | elunii 4914 | . . . 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 4914 | . . 3 ⊢ ((𝐴 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐴 ∈ ∪ ∪ 𝐶) | |
8 | 2, 6, 7 | sylancr 588 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ ∪ ∪ 𝐶) |
9 | 3 | prid2 4768 | . . 3 ⊢ 𝐵 ∈ {𝐴, 𝐵} |
10 | elunii 4914 | . . 3 ⊢ ((𝐵 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ ∪ 𝐶) → 𝐵 ∈ ∪ ∪ 𝐶) | |
11 | 9, 6, 10 | sylancr 588 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐵 ∈ ∪ ∪ 𝐶) |
12 | 8, 11 | jca 513 | 1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 ∈ ∪ ∪ 𝐶 ∧ 𝐵 ∈ ∪ ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3475 {cpr 4631 ⟨cop 4635 ∪ cuni 4909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 |
This theorem is referenced by: asymref 6118 asymref2 6119 wrdexb 14475 |
Copyright terms: Public domain | W3C validator |