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Theorem bj-unrab 33496
Description: Generalization of unrab 4124. 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 3999 . . . 4 𝐴 ⊆ (𝐴𝐵)
2 rabss2 3906 . . . 4 (𝐴 ⊆ (𝐴𝐵) → {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑})
31, 2ax-mp 5 . . 3 {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
4 orc 856 . . . . 5 (𝜑 → (𝜑𝜓))
54a1i 11 . . . 4 (𝑥 ∈ (𝐴𝐵) → (𝜑 → (𝜑𝜓)))
65ss2rabi 3905 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
73, 6sstri 3830 . 2 {𝑥𝐴𝜑} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
8 ssun2 4000 . . . 4 𝐵 ⊆ (𝐴𝐵)
9 rabss2 3906 . . . 4 (𝐵 ⊆ (𝐴𝐵) → {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜓})
108, 9ax-mp 5 . . 3 {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ 𝜓}
11 olc 857 . . . . 5 (𝜓 → (𝜑𝜓))
1211a1i 11 . . . 4 (𝑥 ∈ (𝐴𝐵) → (𝜓 → (𝜑𝜓)))
1312ss2rabi 3905 . . 3 {𝑥 ∈ (𝐴𝐵) ∣ 𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
1410, 13sstri 3830 . 2 {𝑥𝐵𝜓} ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
157, 14unssi 4011 1 ({𝑥𝐴𝜑} ∪ {𝑥𝐵𝜓}) ⊆ {𝑥 ∈ (𝐴𝐵) ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 836  wcel 2107  {crab 3094  cun 3790  wss 3792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rab 3099  df-v 3400  df-un 3797  df-in 3799  df-ss 3806
This theorem is referenced by: (None)
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