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Mirrors > Home > ILE Home > Th. List > uniop | 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 3779 | . . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | unieqi 3821 | . 2 ⊢ ∪ ⟨𝐴, 𝐵⟩ = ∪ {{𝐴}, {𝐴, 𝐵}} |
5 | 1 | snex 4187 | . . 3 ⊢ {𝐴} ∈ V |
6 | prexg 4213 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
7 | 1, 2, 6 | mp2an 426 | . . 3 ⊢ {𝐴, 𝐵} ∈ V |
8 | 5, 7 | unipr 3825 | . 2 ⊢ ∪ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵}) |
9 | snsspr1 3742 | . . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
10 | ssequn1 3307 | . . 3 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}) | |
11 | 9, 10 | mpbi 145 | . 2 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵} |
12 | 4, 8, 11 | 3eqtri 2202 | 1 ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵} |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 Vcvv 2739 ∪ cun 3129 ⊆ wss 3131 {csn 3594 {cpr 3595 ⟨cop 3597 ∪ cuni 3811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 |
This theorem is referenced by: uniopel 4258 elvvuni 4692 dmrnssfld 4892 |
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