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Theorem bj-unrab 33047
Description: Generalization of unrab 3931. 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 3809 . . . 4 𝐴 ⊆ (𝐴𝐵)
2 rabss2 3718 . . . 4 (𝐴 ⊆ (𝐴𝐵) → {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑})
31, 2ax-mp 5 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
4 orc 399 . . . . 5 (𝜑 → (𝜑𝜓))
54a1i 11 . . . 4 (𝑥 ∈ (𝐴𝐵) → (𝜑 → (𝜑𝜓)))
65ss2rabi 3717 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
73, 6sstri 3645 . 2 {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
8 ssun2 3810 . . . 4 𝐵 ⊆ (𝐴𝐵)
9 rabss2 3718 . . . 4 (𝐵 ⊆ (𝐴𝐵) → {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜓})
108, 9ax-mp 5 . . 3 {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜓}
11 olc 398 . . . . 5 (𝜓 → (𝜑𝜓))
1211a1i 11 . . . 4 (𝑥 ∈ (𝐴𝐵) → (𝜓 → (𝜑𝜓)))
1312ss2rabi 3717 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
1410, 13sstri 3645 . 2 {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
157, 14unssi 3821 1 ({𝑥𝐴𝜑} ∪ {𝑥𝐵𝜓}) ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wcel 2030  {crab 2945  cun 3605  wss 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-un 3612  df-in 3614  df-ss 3621
This theorem is referenced by: (None)
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