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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 4343 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ⊆ 𝐴) |
5 | gruss 10211 | . 2 ⊢ ((𝐺 ∈ Univ ∧ 𝐴 ∈ 𝐺 ∧ ∅ ⊆ 𝐴) → ∅ ∈ 𝐺) | |
6 | 1, 2, 4, 5 | syl3anc 1366 | 1 ⊢ (𝜑 → ∅ ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ⊆ wss 3929 ∅c0 4284 Univcgru 10205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-tr 5166 df-iota 6307 df-fv 6356 df-ov 7152 df-gru 10206 |
This theorem is referenced by: grur1cld 40643 grucollcld 40671 |
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