Theorem bnj226 34727

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 3063	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶
3		iunss 5050	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4	2, 3	mpbir 231	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶

Syntax hints:  ∀wral 3059   ⊆ wss 3963  ∪ ciun 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-ss 3980  df-iun 4998

This theorem is referenced by:  bnj229  34877  bnj1128  34983  bnj1145  34986