Theorem bnj226 34710

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 3071	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶
3		iunss 5068	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4	2, 3	mpbir 231	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶

Syntax hints:  ∀wral 3067   ⊆ wss 3976  ∪ ciun 5015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-ss 3993  df-iun 5017

This theorem is referenced by:  bnj229  34860  bnj1128  34966  bnj1145  34969