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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gru0eld | Structured version Visualization version GIF version |
Description: A nonempty Grothendieck universe contains the empty set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
gru0eld.1 | ⊢ (𝜑 → 𝐺 ∈ Univ) |
gru0eld.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐺) |
Ref | Expression |
---|---|
gru0eld | ⊢ (𝜑 → ∅ ∈ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gru0eld.1 | . 2 ⊢ (𝜑 → 𝐺 ∈ Univ) | |
2 | gru0eld.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐺) | |
3 | 0ss 4395 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ⊆ 𝐴) |
5 | gruss 10787 | . 2 ⊢ ((𝐺 ∈ Univ ∧ 𝐴 ∈ 𝐺 ∧ ∅ ⊆ 𝐴) → ∅ ∈ 𝐺) | |
6 | 1, 2, 4, 5 | syl3anc 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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