| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iunxsnf | Structured version Visualization version GIF version | ||
| Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| Ref | Expression |
|---|---|
| iunxsnf.1 | ⊢ Ⅎ𝑥𝐶 |
| iunxsnf.2 | ⊢ 𝐴 ∈ V |
| iunxsnf.3 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| iunxsnf | ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iunxsnf.2 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | iunxsnf.1 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
| 3 | iunxsnf.3 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
| 4 | 2, 3 | iunxsngf 5092 | . 2 ⊢ (𝐴 ∈ V → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) |
| 5 | 1, 4 | ax-mp 5 | 1 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 Vcvv 3480 {csn 4626 ∪ ciun 4991 |
| 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-10 2141 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rex 3071 df-v 3482 df-sbc 3789 df-sn 4627 df-iun 4993 |
| This theorem is referenced by: fiiuncl 45070 iunp1 45071 sge0iunmptlemfi 46428 |
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