Theorem bj-unrab 32922

Description: Generalization of unrab 3898. Equality need not hold. (Contributed by BJ, 21-Apr-2019.)

Assertion

Ref	Expression
bj-unrab	⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ 𝜓}) ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-unrab

Step	Hyp	Ref	Expression
1		ssun1 3776	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		rabss2 3685	. . . 4 ⊢ (𝐴 ⊆ (𝐴 ∪ 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜑})
3	1, 2	ax-mp 5	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜑}
4		orc 400	. . . . 5 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
5	4	a1i 11	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → (𝜑 → (𝜑 ∨ 𝜓)))
6	5	ss2rabi 3684	. . 3 ⊢ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)}
7	3, 6	sstri 3612	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)}
8		ssun2 3777	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
9		rabss2 3685	. . . 4 ⊢ (𝐵 ⊆ (𝐴 ∪ 𝐵) → {𝑥 ∈ 𝐵 ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜓})
10	8, 9	ax-mp 5	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜓}
11		olc 399	. . . . 5 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
12	11	a1i 11	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → (𝜓 → (𝜑 ∨ 𝜓)))
13	12	ss2rabi 3684	. . 3 ⊢ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)}
14	10, 13	sstri 3612	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)}
15	7, 14	unssi 3788	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ 𝜓}) ⊆ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ (𝜑 ∨ 𝜓)}

Syntax hints:   → wi 4   ∨ wo 383   ∈ wcel 1990  {crab 2916   ∪ cun 3572   ⊆ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-un 3579  df-in 3581  df-ss 3588

This theorem is referenced by: (None)