Theorem uniss 1936

Description: Subclass relationship for class union. Theorem 61 of [Suppes] p. 39.

Assertion

Ref	Expression
uniss	⊢ (A ⊆ B → ∪A ⊆ ∪B)

Proof of Theorem uniss

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . 6 ⊢ (A ⊆ B → (y ∈ A → y ∈ B))
2	1	anim2d 433	. . . . 5 ⊢ (A ⊆ B → ((x ∈ y ∧ y ∈ A) → (x ∈ y ∧ y ∈ B)))
3	2	19.22dv 947	. . . 4 ⊢ (A ⊆ B → (∃y(x ∈ y ∧ y ∈ A) → ∃y(x ∈ y ∧ y ∈ B)))
4	3	19.21aiv 943	. . 3 ⊢ (A ⊆ B → ∀x(∃y(x ∈ y ∧ y ∈ A) → ∃y(x ∈ y ∧ y ∈ B)))
5		ss2ab 1551	. . 3 ⊢ ({x∣∃y(x ∈ y ∧ y ∈ A)} ⊆ {x∣∃y(x ∈ y ∧ y ∈ B)} ↔ ∀x(∃y(x ∈ y ∧ y ∈ A) → ∃y(x ∈ y ∧ y ∈ B)))
6	4, 5	sylibr 175	. 2 ⊢ (A ⊆ B → {x∣∃y(x ∈ y ∧ y ∈ A)} ⊆ {x∣∃y(x ∈ y ∧ y ∈ B)})
7		df-uni 1920	. 2 ⊢ ∪A = {x∣∃y(x ∈ y ∧ y ∈ A)}
8		df-uni 1920	. 2 ⊢ ∪B = {x∣∃y(x ∈ y ∧ y ∈ B)}
9	6, 7, 8	3sstr4g 1541	1 ⊢ (A ⊆ B → ∪A ⊆ ∪B)

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803  {cab 1090   ∈ wcel 1092   ⊆ wss 1487  ∪cuni 1919

This theorem is referenced by:  uni0 1938  unidif 1943  trcl 3489  cflim 3704  dfchsup2 5299  hsupval2t 5301  hsupvalt 5302  shsupclt 5307  hsupss 5310  shsupunss 5316

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-in 1491  df-ss 1492  df-uni 1920

metamath.org