Theorem uniss 3601

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 2939	. . . . 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 1760	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
4		eluni 3583	. . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
5		eluni 3583	. . 3 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
6	3, 4, 5	3imtr4g 194	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ ∪ 𝐵))
7	6	ssrdv 2951	1 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1381   ∈ wcel 1393   ⊆ wss 2917  ∪ cuni 3580

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581

This theorem is referenced by:  unissi  3603  unissd  3604  intssuni2m  3639  relfld  4846