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Theorem bj-unrab 36927
Description: Generalization of unrab 4315. Equality need not hold. (Contributed by BJ, 21-Apr-2019.)
Assertion
Ref Expression
bj-unrab ({𝑥𝐴𝜑} ∪ {𝑥𝐵𝜓}) ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-unrab
StepHypRef Expression
1 ssun1 4178 . . . 4 𝐴 ⊆ (𝐴𝐵)
2 rabss2 4078 . . . 4 (𝐴 ⊆ (𝐴𝐵) → {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑})
31, 2ax-mp 5 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
4 orc 868 . . . . 5 (𝜑 → (𝜑𝜓))
54a1i 11 . . . 4 (𝑥 ∈ (𝐴𝐵) → (𝜑 → (𝜑𝜓)))
65ss2rabi 4077 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
73, 6sstri 3993 . 2 {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
8 ssun2 4179 . . . 4 𝐵 ⊆ (𝐴𝐵)
9 rabss2 4078 . . . 4 (𝐵 ⊆ (𝐴𝐵) → {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜓})
108, 9ax-mp 5 . . 3 {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜓}
11 olc 869 . . . . 5 (𝜓 → (𝜑𝜓))
1211a1i 11 . . . 4 (𝑥 ∈ (𝐴𝐵) → (𝜓 → (𝜑𝜓)))
1312ss2rabi 4077 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
1410, 13sstri 3993 . 2 {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
157, 14unssi 4191 1 ({𝑥𝐴𝜑} ∪ {𝑥𝐵𝜓}) ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 848  wcel 2108  {crab 3436  cun 3949  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-v 3482  df-un 3956  df-ss 3968
This theorem is referenced by: (None)
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