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Theorem gru0eld 42921
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 4395 . . 3 ∅ ⊆ 𝐴
43a1i 11 . 2 (𝜑 → ∅ ⊆ 𝐴)
5 gruss 10787 . 2 ((𝐺 ∈ Univ ∧ 𝐴𝐺 ∧ ∅ ⊆ 𝐴) → ∅ ∈ 𝐺)
61, 2, 4, 5syl3anc 1372 1 (𝜑 → ∅ ∈ 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wss 3947  c0 4321  Univcgru 10781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-tr 5265  df-iota 6492  df-fv 6548  df-ov 7407  df-gru 10782
This theorem is referenced by:  grur1cld  42924  grucollcld  42952
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