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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 4795 | . . 3 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | unieqi 4840 | . 2 ⊢ ∪ 〈𝐴, 𝐵〉 = ∪ {{𝐴}, {𝐴, 𝐵}} |
5 | snex 5323 | . . 3 ⊢ {𝐴} ∈ V | |
6 | prex 5324 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
7 | 5, 6 | unipr 4844 | . 2 ⊢ ∪ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵}) |
8 | snsspr1 4740 | . . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
9 | ssequn1 4155 | . . 3 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}) | |
10 | 8, 9 | mpbi 232 | . 2 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵} |
11 | 4, 7, 10 | 3eqtri 2848 | 1 ⊢ ∪ 〈𝐴, 𝐵〉 = {𝐴, 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∪ cun 3933 ⊆ wss 3935 {csn 4560 {cpr 4562 〈cop 4566 ∪ cuni 4831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 |
This theorem is referenced by: uniopel 5398 elvvuni 5622 dmrnssfld 5835 dffv2 6750 rankxplim 9302 |
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