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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 4872 | . . 3 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
| 4 | 3 | unieqi 4919 | . 2 ⊢ ∪ 〈𝐴, 𝐵〉 = ∪ {{𝐴}, {𝐴, 𝐵}} |
| 5 | snex 5436 | . . 3 ⊢ {𝐴} ∈ V | |
| 6 | prex 5437 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
| 7 | 5, 6 | unipr 4924 | . 2 ⊢ ∪ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵}) |
| 8 | snsspr1 4814 | . . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
| 9 | ssequn1 4186 | . . 3 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}) | |
| 10 | 8, 9 | mpbi 230 | . 2 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵} |
| 11 | 4, 7, 10 | 3eqtri 2769 | 1 ⊢ ∪ 〈𝐴, 𝐵〉 = {𝐴, 𝐵} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 {csn 4626 {cpr 4628 〈cop 4632 ∪ cuni 4907 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 |
| This theorem is referenced by: uniopel 5521 elvvuni 5762 dmrnssfld 5984 dffv2 7004 rankxplim 9919 |
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