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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 4824 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
4 | 1, 2, 3 | syl2anc 693 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∅c0 3915 ∩ ciin 4521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-int 4476 df-iin 4523 |
This theorem is referenced by: smfsuplem1 41017 smfinflem 41023 smflimsuplem1 41026 smflimsuplem2 41027 smflimsuplem3 41028 smflimsuplem4 41029 smflimsuplem5 41030 smflimsuplem7 41032 |
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