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Mirrors > Home > MPE Home > Th. List > iunxsn | Structured version Visualization version GIF version |
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.) |
Ref | Expression |
---|---|
iunxsn.1 | ⊢ 𝐴 ∈ V |
iunxsn.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iunxsn | ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | iunxsn.2 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
3 | 2 | iunxsng 5094 | . 2 ⊢ (𝐴 ∈ V → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 {csn 4630 ∪ ciun 4997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-v 3463 df-sn 4631 df-iun 4999 |
This theorem is referenced by: iunsuc 6456 funopsn 7157 fparlem3 8119 fparlem4 8120 iunfi 9366 kmlem11 10185 ackbij1lem8 10252 dfid6 15011 fsum2dlem 15752 fsumiun 15803 fprod2dlem 15960 prmreclem4 16891 fiuncmp 23352 ovolfiniun 25474 finiunmbl 25517 volfiniun 25520 voliunlem1 25523 iuninc 32430 cvmliftlem10 35035 mrsubvrs 35263 dfrcl4 43248 iunrelexp0 43274 corclrcl 43279 cotrcltrcl 43297 trclfvdecomr 43300 dfrtrcl4 43310 corcltrcl 43311 cotrclrcl 43314 |
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