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Theorem bj-unrab 36445
Description: Generalization of unrab 4308. 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 4174 . . . 4 𝐴 ⊆ (𝐴𝐵)
2 rabss2 4075 . . . 4 (𝐴 ⊆ (𝐴𝐵) → {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑})
31, 2ax-mp 5 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
4 orc 865 . . . . 5 (𝜑 → (𝜑𝜓))
54a1i 11 . . . 4 (𝑥 ∈ (𝐴𝐵) → (𝜑 → (𝜑𝜓)))
65ss2rabi 4074 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
73, 6sstri 3991 . 2 {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
8 ssun2 4175 . . . 4 𝐵 ⊆ (𝐴𝐵)
9 rabss2 4075 . . . 4 (𝐵 ⊆ (𝐴𝐵) → {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜓})
108, 9ax-mp 5 . . 3 {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜓}
11 olc 866 . . . . 5 (𝜓 → (𝜑𝜓))
1211a1i 11 . . . 4 (𝑥 ∈ (𝐴𝐵) → (𝜓 → (𝜑𝜓)))
1312ss2rabi 4074 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
1410, 13sstri 3991 . 2 {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
157, 14unssi 4187 1 ({𝑥𝐴𝜑} ∪ {𝑥𝐵𝜓}) ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 845  wcel 2098  {crab 3430  cun 3947  wss 3949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rab 3431  df-v 3475  df-un 3954  df-in 3956  df-ss 3966
This theorem is referenced by: (None)
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