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Mirrors > Home > MPE Home > Th. List > uniop | Structured version Visualization version GIF version |
Description: The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opthw.1 | ⊢ 𝐴 ∈ V |
opthw.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
uniop | ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opthw.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | opthw.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | dfop 4865 | . . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | unieqi 4912 | . 2 ⊢ ∪ ⟨𝐴, 𝐵⟩ = ∪ {{𝐴}, {𝐴, 𝐵}} |
5 | snex 5422 | . . 3 ⊢ {𝐴} ∈ V | |
6 | prex 5423 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
7 | 5, 6 | unipr 4917 | . 2 ⊢ ∪ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵}) |
8 | snsspr1 4810 | . . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
9 | ssequn1 4173 | . . 3 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}) | |
10 | 8, 9 | mpbi 229 | . 2 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵} |
11 | 4, 7, 10 | 3eqtri 2756 | 1 ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∪ cun 3939 ⊆ wss 3941 {csn 4621 {cpr 4623 ⟨cop 4627 ∪ cuni 4900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 |
This theorem is referenced by: uniopel 5507 elvvuni 5743 dmrnssfld 5960 dffv2 6977 rankxplim 9871 |
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