Theorem uniss 3885

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 3195	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵))
2	1	anim2d 337	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
3	2	eximdv 1904	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
4		eluni 3867	. . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
5		eluni 3867	. . 3 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
6	3, 4, 5	3imtr4g 205	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ ∪ 𝐵))
7	6	ssrdv 3207	1 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1516   ∈ wcel 2178   ⊆ wss 3174  ∪ cuni 3864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865

This theorem is referenced by:  unissi  3887  unissd  3888  intssuni2m  3923  relfld  5230  prdsvallem  13219  prdsval  13220  tgcl  14651  distop  14672