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Theorem bj-unrab 31975
Description: Generalization of unrab 3760. 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 3642 . . . 4 𝐴 ⊆ (𝐴𝐵)
2 rabss2 3552 . . . 4 (𝐴 ⊆ (𝐴𝐵) → {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑})
31, 2ax-mp 5 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
4 orc 398 . . . . 5 (𝜑 → (𝜑𝜓))
54a1i 11 . . . 4 (𝑥 ∈ (𝐴𝐵) → (𝜑 → (𝜑𝜓)))
65ss2rabi 3551 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
73, 6sstri 3481 . 2 {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
8 ssun2 3643 . . . 4 𝐵 ⊆ (𝐴𝐵)
9 rabss2 3552 . . . 4 (𝐵 ⊆ (𝐴𝐵) → {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜓})
108, 9ax-mp 5 . . 3 {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜓}
11 olc 397 . . . . 5 (𝜓 → (𝜑𝜓))
1211a1i 11 . . . 4 (𝑥 ∈ (𝐴𝐵) → (𝜓 → (𝜑𝜓)))
1312ss2rabi 3551 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
1410, 13sstri 3481 . 2 {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
157, 14unssi 3654 1 ({𝑥𝐴𝜑} ∪ {𝑥𝐵𝜓}) ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  wcel 1938  {crab 2804  cun 3442  wss 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rab 2809  df-v 3079  df-un 3449  df-in 3451  df-ss 3458
This theorem is referenced by: (None)
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