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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 3539 | . . 3 ⊢ 〈A, B〉 = {{A}, {A, B}} |
4 | 3 | unieqi 3581 | . 2 ⊢ ∪ 〈A, B〉 = ∪ {{A}, {A, B}} |
5 | snexgOLD 3926 | . . . 4 ⊢ (A ∈ V → {A} ∈ V) | |
6 | 1, 5 | ax-mp 7 | . . 3 ⊢ {A} ∈ V |
7 | prexgOLD 3937 | . . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → {A, B} ∈ V) | |
8 | 1, 2, 7 | mp2an 402 | . . 3 ⊢ {A, B} ∈ V |
9 | 6, 8 | unipr 3585 | . 2 ⊢ ∪ {{A}, {A, B}} = ({A} ∪ {A, B}) |
10 | snsspr1 3503 | . . 3 ⊢ {A} ⊆ {A, B} | |
11 | ssequn1 3107 | . . 3 ⊢ ({A} ⊆ {A, B} ↔ ({A} ∪ {A, B}) = {A, B}) | |
12 | 10, 11 | mpbi 133 | . 2 ⊢ ({A} ∪ {A, B}) = {A, B} |
13 | 4, 9, 12 | 3eqtri 2061 | 1 ⊢ ∪ 〈A, B〉 = {A, B} |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∈ wcel 1390 Vcvv 2551 ∪ cun 2909 ⊆ wss 2911 {csn 3367 {cpr 3368 〈cop 3370 ∪ cuni 3571 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 |
This theorem is referenced by: uniopel 3984 elvvuni 4347 dmrnssfld 4538 |
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