Theorem bnj226 30056

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 2908	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶
3		iunss 4497	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4	2, 3	mpbir 220	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶

Syntax hints:  ∀wral 2896   ⊆ wss 3540  ∪ ciun 4455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-iun 4457

This theorem is referenced by:  bnj229  30208  bnj1128  30312  bnj1145  30315