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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 3674 | . . 3 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | unieqi 3716 | . 2 ⊢ ∪ 〈𝐴, 𝐵〉 = ∪ {{𝐴}, {𝐴, 𝐵}} |
5 | 1 | snex 4079 | . . 3 ⊢ {𝐴} ∈ V |
6 | prexg 4103 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
7 | 1, 2, 6 | mp2an 422 | . . 3 ⊢ {𝐴, 𝐵} ∈ V |
8 | 5, 7 | unipr 3720 | . 2 ⊢ ∪ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵}) |
9 | snsspr1 3638 | . . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
10 | ssequn1 3216 | . . 3 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}) | |
11 | 9, 10 | mpbi 144 | . 2 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵} |
12 | 4, 8, 11 | 3eqtri 2142 | 1 ⊢ ∪ 〈𝐴, 𝐵〉 = {𝐴, 𝐵} |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∈ wcel 1465 Vcvv 2660 ∪ cun 3039 ⊆ wss 3041 {csn 3497 {cpr 3498 〈cop 3500 ∪ cuni 3706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 |
This theorem is referenced by: uniopel 4148 elvvuni 4573 dmrnssfld 4772 |
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