Theorem bnj226 35432

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 3084	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶
3		iunss 5006	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4	2, 3	mpbir 234	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶

This proof depends on syntax axioms:  ∀wral 3080   ⊆ wss 3900  ∪ ciun 4954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-ss 3901  df-iun 4956

This theorem is used by:  bnj229  35581  bnj1128  35687  bnj1145  35690