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Theorem gru0eld 43545
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 4391 . . 3 ∅ ⊆ 𝐴
43a1i 11 . 2 (𝜑 → ∅ ⊆ 𝐴)
5 gruss 10790 . 2 ((𝐺 ∈ Univ ∧ 𝐴𝐺 ∧ ∅ ⊆ 𝐴) → ∅ ∈ 𝐺)
61, 2, 4, 5syl3anc 1368 1 (𝜑 → ∅ ∈ 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wss 3943  c0 4317  Univcgru 10784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-tr 5259  df-iota 6488  df-fv 6544  df-ov 7407  df-gru 10785
This theorem is referenced by:  grur1cld  43548  grucollcld  43576
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