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 5016 | . 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 1537 ∈ wcel 2114 Ⅎwnfc 2963 Vcvv 3496 {csn 4569 ∪ ciun 4921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-sbc 3775 df-sn 4570 df-iun 4923 |
This theorem is referenced by: fiiuncl 41334 iunp1 41335 sge0iunmptlemfi 42702 |
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