Theorem iunn0m 4031

Description: There is an inhabited class in an indexed collection 𝐵(𝑥) iff the indexed union of them is inhabited. (Contributed by Jim Kingdon, 16-Aug-2018.)

Assertion

Ref	Expression
iunn0m	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunn0m

Step	Hyp	Ref	Expression
1		rexcom4 2826	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		eliun 3974	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	exbii 1653	. 2 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4	1, 3	bitr4i 187	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   ↔ wb 105  ∃wex 1540   ∈ wcel 2202  ∃wrex 2511  ∪ ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-iun 3972

This theorem is referenced by: (None)