Theorem ssuni 3955

Description: Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
ssuni	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶)

Proof of Theorem ssuni

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

Step	Hyp	Ref	Expression
1		eleq2 2302	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐵))
2	1	imbi1d 231	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 → 𝑦 ∈ ∪ 𝐶)))
3		elunii 3938	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ ∪ 𝐶)
4	3	expcom 116	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝐶))
5	2, 4	vtoclga 2889	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ ∪ 𝐶))
6	5	imim2d 54	. . . 4 ⊢ (𝐵 ∈ 𝐶 → ((𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐴 → 𝑦 ∈ ∪ 𝐶)))
7	6	alimdv 1932	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ ∪ 𝐶)))
8		ssalel 3235	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵))
9		ssalel 3235	. . 3 ⊢ (𝐴 ⊆ ∪ 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ ∪ 𝐶))
10	7, 8, 9	3imtr4g 205	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ⊆ ∪ 𝐶))
11	10	impcom 125	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1400   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220  ∪ cuni 3933

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-uni 3934

This theorem is referenced by:  elssuni  3961  uniss2  3964  ssorduni  4632