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Theorem gru0eld 42978
Description: A nonempty Grothendieck universe contains the empty set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypotheses
Ref Expression
gru0eld.1 (𝜑𝐺 ∈ Univ)
gru0eld.2 (𝜑𝐴𝐺)
Assertion
Ref Expression
gru0eld (𝜑 → ∅ ∈ 𝐺)

Proof of Theorem gru0eld
StepHypRef Expression
1 gru0eld.1 . 2 (𝜑𝐺 ∈ Univ)
2 gru0eld.2 . 2 (𝜑𝐴𝐺)
3 0ss 4396 . . 3 ∅ ⊆ 𝐴
43a1i 11 . 2 (𝜑 → ∅ ⊆ 𝐴)
5 gruss 10790 . 2 ((𝐺 ∈ Univ ∧ 𝐴𝐺 ∧ ∅ ⊆ 𝐴) → ∅ ∈ 𝐺)
61, 2, 4, 5syl3anc 1371 1 (𝜑 → ∅ ∈ 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wss 3948  c0 4322  Univcgru 10784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  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-tr 5266  df-iota 6495  df-fv 6551  df-ov 7411  df-gru 10785
This theorem is referenced by:  grur1cld  42981  grucollcld  43009
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