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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 5354 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 Vcvv 3478 ∅c0 4339 ∩ ciin 4997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 df-nul 4340 df-int 4952 df-iin 4999 |
This theorem is referenced by: smfsuplem1 46767 smfinflem 46773 smflimsuplem1 46776 smflimsuplem2 46777 smflimsuplem3 46778 smflimsuplem4 46779 smflimsuplem5 46780 smflimsuplem7 46782 |
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