Theorem ssiun2 2019

Description: Identity law for subset of an indexed union.

Assertion

Ref	Expression
ssiun2	⊢ (x ∈ A → B ⊆ ∪x ∈ A B)

Proof of Theorem ssiun2

Step	Hyp	Ref	Expression
1		ra4e 1244	. . . 4 ⊢ ((x ∈ A ∧ y ∈ B) → ∃x ∈ A y ∈ B)
2		eliun 1998	. . . 4 ⊢ (y ∈ ∪x ∈ A B ↔ ∃x ∈ A y ∈ B)
3	1, 2	sylibr 175	. . 3 ⊢ ((x ∈ A ∧ y ∈ B) → y ∈ ∪x ∈ A B)
4	3	exp 291	. 2 ⊢ (x ∈ A → (y ∈ B → y ∈ ∪x ∈ A B))
5	4	ssrdv 1509	1 ⊢ (x ∈ A → B ⊆ ∪x ∈ A B)

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  ∃wrex 1202   ⊆ wss 1487  ∪ciun 1994

This theorem is referenced by:  ssiun2s 2020  r1val1 3502  rankuni 3533  ranklon 3540  cplem1 3545  infxpidmlem5 4937

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-in 1491  df-ss 1492  df-iun 1996

metamath.org