Theorem unss1 2196

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		pm2.38 568	. . . 4 ⊢ ((x ∈ A → x ∈ B) → ((x ∈ A ⋁ x ∈ C) → (x ∈ B ⋁ x ∈ C)))
2		elun 2170	. . . 4 ⊢ (x ∈ (A ∪ C) ↔ (x ∈ A ⋁ x ∈ C))
3		elun 2170	. . . 4 ⊢ (x ∈ (B ∪ C) ↔ (x ∈ B ⋁ x ∈ C))
4	1, 2, 3	3imtr4g 552	. . 3 ⊢ ((x ∈ A → x ∈ B) → (x ∈ (A ∪ C) → x ∈ (B ∪ C)))
5	4	19.20i 991	. 2 ⊢ (∀x(x ∈ A → x ∈ B) → ∀x(x ∈ (A ∪ C) → x ∈ (B ∪ C)))
6		dfss2 2055	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
7		dfss2 2055	. 2 ⊢ ((A ∪ C) ⊆ (B ∪ C) ↔ ∀x(x ∈ (A ∪ C) → x ∈ (B ∪ C)))
8	5, 6, 7	3imtr4 219	1 ⊢ (A ⊆ B → (A ∪ C) ⊆ (B ∪ C))

Syntax hints:   → wi 3   ⋁ wo 222  ∀wal 953   ∈ wcel 957   ∪ cun 2042   ⊆ wss 2044

This theorem is referenced by:  unss2 2198  unss12 2199  eldifpw 2906

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-un 2047  df-in 2048  df-ss 2050

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