Theorem unss1 1627

Description: Subclass law for union of classes.

Assertion

Ref	Expression
unss1	⊢ (A ⊆ B → (A ∪ C) ⊆ (B ∪ C))

Proof of Theorem unss1

Step	Hyp	Ref	Expression
1		id 9	. . . . 5 ⊢ ((x ∈ A → x ∈ B) → (x ∈ A → x ∈ B))
2	1	orim1d 437	. . . 4 ⊢ ((x ∈ A → x ∈ B) → ((x ∈ A ∨ x ∈ C) → (x ∈ B ∨ x ∈ C)))
3		elun 1601	. . . 4 ⊢ (x ∈ (A ∪ C) ↔ (x ∈ A ∨ x ∈ C))
4		elun 1601	. . . 4 ⊢ (x ∈ (B ∪ C) ↔ (x ∈ B ∨ x ∈ C))
5	2, 3, 4	3imtr4g 426	. . 3 ⊢ ((x ∈ A → x ∈ B) → (x ∈ (A ∪ C) → x ∈ (B ∪ C)))
6	5	19.20i 691	. 2 ⊢ (∀x(x ∈ A → x ∈ B) → ∀x(x ∈ (A ∪ C) → x ∈ (B ∪ C)))
7		dfss2 1497	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
8		dfss2 1497	. 2 ⊢ ((A ∪ C) ⊆ (B ∪ C) ↔ ∀x(x ∈ (A ∪ C) → x ∈ (B ∪ C)))
9	6, 7, 8	3imtr4 192	1 ⊢ (A ⊆ B → (A ∪ C) ⊆ (B ∪ C))

Syntax hints:   → wi 2   ∨ wo 195  ∀wal 672   ∈ wcel 1092   ∪ cun 1485   ⊆ wss 1487

This theorem is referenced by:  unss2 1629  unss12 1630

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-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492

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