Theorem ssuniOLD 4396

Description: Obsolete proof of ssuni 4395 as of 26-Jul-2021. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ssuniOLD

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

Step	Hyp	Ref	Expression
1		eleq2 2677	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐵))
2	1	imbi1d 330	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 → 𝑦 ∈ ∪ 𝐶)))
3		elunii 4377	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ ∪ 𝐶)
4	3	expcom 450	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝐶))
5	2, 4	vtoclga 3245	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ ∪ 𝐶))
6	5	imim2d 55	. . . 4 ⊢ (𝐵 ∈ 𝐶 → ((𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐴 → 𝑦 ∈ ∪ 𝐶)))
7	6	alimdv 1832	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ ∪ 𝐶)))
8		dfss2 3557	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵))
9		dfss2 3557	. . 3 ⊢ (𝐴 ⊆ ∪ 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ ∪ 𝐶))
10	7, 8, 9	3imtr4g 284	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ⊆ ∪ 𝐶))
11	10	impcom 445	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373

This theorem is referenced by: (None)