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Theorem pgindlem 47925
Description: Lemma for pgind 47927. (Contributed by Emmett Weisz, 27-May-2024.) (New usage is discouraged.)
Assertion
Ref Expression
pgindlem (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st𝑥) ∪ (2nd𝑥)) ⊆ 𝑧)

Proof of Theorem pgindlem
StepHypRef Expression
1 xp1st 8011 . . 3 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (1st𝑥) ∈ 𝒫 𝑧)
21elpwid 4611 . 2 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (1st𝑥) ⊆ 𝑧)
3 xp2nd 8012 . . 3 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (2nd𝑥) ∈ 𝒫 𝑧)
43elpwid 4611 . 2 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (2nd𝑥) ⊆ 𝑧)
52, 4unssd 4186 1 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st𝑥) ∪ (2nd𝑥)) ⊆ 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cun 3946  wss 3948  𝒫 cpw 4602   × cxp 5674  cfv 6543  1st c1st 7977  2nd c2nd 7978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7979  df-2nd 7980
This theorem is referenced by:  pgindnf  47926
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