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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 4878 | . . 3 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | unieqi 4925 | . 2 ⊢ ∪ 〈𝐴, 𝐵〉 = ∪ {{𝐴}, {𝐴, 𝐵}} |
5 | snex 5437 | . . 3 ⊢ {𝐴} ∈ V | |
6 | prex 5438 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
7 | 5, 6 | unipr 4930 | . 2 ⊢ ∪ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵}) |
8 | snsspr1 4823 | . . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
9 | ssequn1 4181 | . . 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 2758 | 1 ⊢ ∪ 〈𝐴, 𝐵〉 = {𝐴, 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∪ cun 3945 ⊆ wss 3947 {csn 4633 {cpr 4635 〈cop 4639 ∪ cuni 4913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 |
This theorem is referenced by: uniopel 5522 elvvuni 5758 dmrnssfld 5977 dffv2 6997 rankxplim 9922 |
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