Theorem bnj226 34748

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj226.1	⊢ 𝐵 ⊆ 𝐶

Assertion

Ref	Expression
bnj226	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem bnj226

Step	Hyp	Ref	Expression
1		bnj226.1	. . 3 ⊢ 𝐵 ⊆ 𝐶
2	1	rgenw 3065	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶
3		iunss 5045	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4	2, 3	mpbir 231	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶

Syntax hints:  ∀wral 3061   ⊆ wss 3951  ∪ ciun 4991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-ss 3968  df-iun 4993

This theorem is referenced by:  bnj229  34898  bnj1128  35004  bnj1145  35007