Theorem uniss 3592

Description: Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
uniss	⊢ (A ⊆ B → ∪ A ⊆ ∪ B)

Proof of Theorem uniss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 2933	. . . . 5 ⊢ (A ⊆ B → (y ∈ A → y ∈ B))
2	1	anim2d 320	. . . 4 ⊢ (A ⊆ B → ((x ∈ y ∧ y ∈ A) → (x ∈ y ∧ y ∈ B)))
3	2	eximdv 1757	. . 3 ⊢ (A ⊆ B → (∃y(x ∈ y ∧ y ∈ A) → ∃y(x ∈ y ∧ y ∈ B)))
4		eluni 3574	. . 3 ⊢ (x ∈ ∪ A ↔ ∃y(x ∈ y ∧ y ∈ A))
5		eluni 3574	. . 3 ⊢ (x ∈ ∪ B ↔ ∃y(x ∈ y ∧ y ∈ B))
6	3, 4, 5	3imtr4g 194	. 2 ⊢ (A ⊆ B → (x ∈ ∪ A → x ∈ ∪ B))
7	6	ssrdv 2945	1 ⊢ (A ⊆ B → ∪ A ⊆ ∪ B)

Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911  ∪ cuni 3571

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-bnd 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-v 2553  df-in 2918  df-ss 2925  df-uni 3572

This theorem is referenced by:  unissi  3594  unissd  3595  intssuni2m  3630  relfld  4789