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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iinexd | Structured version Visualization version GIF version |
Description: The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
iinexd.1 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
iinexd.2 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) |
Ref | Expression |
---|---|
iinexd | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iinexd.1 | . 2 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
2 | iinexd.2 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) | |
3 | iinexg 5332 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
4 | 1, 2, 3 | syl2anc 583 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 Vcvv 3466 ∅c0 4315 ∩ ciin 4989 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-in 3948 df-ss 3958 df-nul 4316 df-int 4942 df-iin 4991 |
This theorem is referenced by: smfsuplem1 46073 smfinflem 46079 smflimsuplem1 46082 smflimsuplem2 46083 smflimsuplem3 46084 smflimsuplem4 46085 smflimsuplem5 46086 smflimsuplem7 46088 |
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