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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 4339 | . . 3 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | unieqi 4381 | . 2 ⊢ ∪ 〈𝐴, 𝐵〉 = ∪ {{𝐴}, {𝐴, 𝐵}} |
5 | snex 4835 | . . 3 ⊢ {𝐴} ∈ V | |
6 | prex 4836 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
7 | 5, 6 | unipr 4385 | . 2 ⊢ ∪ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵}) |
8 | snsspr1 4285 | . . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
9 | ssequn1 3745 | . . 3 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}) | |
10 | 8, 9 | mpbi 219 | . 2 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵} |
11 | 4, 7, 10 | 3eqtri 2636 | 1 ⊢ ∪ 〈𝐴, 𝐵〉 = {𝐴, 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 {csn 4125 {cpr 4127 〈cop 4131 ∪ cuni 4372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 |
This theorem is referenced by: uniopel 4903 elvvuni 5102 dmrnssfld 5305 dffv2 6181 rankxplim 8625 |
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