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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 4282 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ⊆ 𝐴) |
5 | gruss 10289 | . 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 2113 ⊆ wss 3841 ∅c0 4209 Univcgru 10283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-tr 5134 df-iota 6291 df-fv 6341 df-ov 7167 df-gru 10284 |
This theorem is referenced by: grur1cld 41376 grucollcld 41404 |
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