iunn0 - New Foundations Explorer

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Theorem iunn0 4026

Description: There is a non-empty class in an indexed collection B(x) iff the indexed union of them is non-empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

**Assertion**
Ref	Expression
iunn0	⊢ (∃x ∈ A B ≠ ∅ ↔ ∪x ∈ A B ≠ ∅)

Distinct variable group: x,A Allowed substitution hint: B(x)

Proof of Theorem iunn0 Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom4 2878	. . 3 ⊢ (∃x ∈ A ∃y y ∈ B ↔ ∃y∃x ∈ A y ∈ B)
2		eliun 3973	. . . 4 ⊢ (y ∈ ∪x ∈ A B ↔ ∃x ∈ A y ∈ B)
3	2	exbii 1582	. . 3 ⊢ (∃y y ∈ ∪x ∈ A B ↔ ∃y∃x ∈ A y ∈ B)
4	1, 3	bitr4i 243	. 2 ⊢ (∃x ∈ A ∃y y ∈ B ↔ ∃y y ∈ ∪x ∈ A B)
5		n0 3559	. . 3 ⊢ (B ≠ ∅ ↔ ∃y y ∈ B)
6	5	rexbii 2639	. 2 ⊢ (∃x ∈ A B ≠ ∅ ↔ ∃x ∈ A ∃y y ∈ B)
7		n0 3559	. 2 ⊢ (∪x ∈ A B ≠ ∅ ↔ ∃y y ∈ ∪x ∈ A B)
8	4, 6, 7	3bitr4i 268	1 ⊢ (∃x ∈ A B ≠ ∅ ↔ ∪x ∈ A B ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 176 ∃wex 1541 ∈ wcel 1710 ≠ wne 2516 ∃wrex 2615 ∅c0 3550 ∪ciun 3969

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 df-iun 3971

This theorem is referenced by: (None)

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