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Theorem gru0eld 44253
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 4399 . . 3 ∅ ⊆ 𝐴
43a1i 11 . 2 (𝜑 → ∅ ⊆ 𝐴)
5 gruss 10837 . 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 3950  c0 4332  Univcgru 10831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-tr 5259  df-iota 6513  df-fv 6568  df-ov 7435  df-gru 10832
This theorem is referenced by:  grur1cld  44256  grucollcld  44284
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