Theorem iunss1 3659

Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iunss1	⊢ (A ⊆ B → ∪ x ∈ A 𝐶 ⊆ ∪ x ∈ B 𝐶)

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   𝐶(x)

Proof of Theorem iunss1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrexv 2999	. . 3 ⊢ (A ⊆ B → (∃x ∈ A y ∈ 𝐶 → ∃x ∈ B y ∈ 𝐶))
2		eliun 3652	. . 3 ⊢ (y ∈ ∪ x ∈ A 𝐶 ↔ ∃x ∈ A y ∈ 𝐶)
3		eliun 3652	. . 3 ⊢ (y ∈ ∪ x ∈ B 𝐶 ↔ ∃x ∈ B y ∈ 𝐶)
4	1, 2, 3	3imtr4g 194	. 2 ⊢ (A ⊆ B → (y ∈ ∪ x ∈ A 𝐶 → y ∈ ∪ x ∈ B 𝐶))
5	4	ssrdv 2945	1 ⊢ (A ⊆ B → ∪ x ∈ A 𝐶 ⊆ ∪ x ∈ B 𝐶)

Syntax hints:   → wi 4   ∈ wcel 1390  ∃wrex 2301   ⊆ wss 2911  ∪ ciun 3648

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iun 3650

This theorem is referenced by:  iuneq1  3661  iunxdif2  3696