Theorem iunn0 4027

Description: There is a nonempty class in an indexed collection B(x) iff the indexed union of them is nonempty. (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 2879	. . 3 ⊢ (∃x ∈ A ∃y y ∈ B ↔ ∃y∃x ∈ A y ∈ B)
2		eliun 3974	. . . 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 3560	. . 3 ⊢ (B ≠ ∅ ↔ ∃y y ∈ B)
6	5	rexbii 2640	. 2 ⊢ (∃x ∈ A B ≠ ∅ ↔ ∃x ∈ A ∃y y ∈ B)
7		n0 3560	. 2 ⊢ (∪x ∈ A B ≠ ∅ ↔ ∃y y ∈ ∪x ∈ A B)
8	4, 6, 7	3bitr4i 268	1 ⊢ (∃x ∈ A B ≠ ∅ ↔ ∪x ∈ A B ≠ ∅)

Syntax hints:   ↔ wb 176  ∃wex 1541   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616  ∅c0 3551  ∪ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-iun 3972

This theorem is referenced by: (None)