![]() |
Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > grusucd | Structured version Visualization version GIF version |
Description: Grothendieck universes are closed under ordinal successor. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
Ref | Expression |
---|---|
grusucd.1 | ⊢ (𝜑 → 𝐺 ∈ Univ) |
grusucd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐺) |
Ref | Expression |
---|---|
grusucd | ⊢ (𝜑 → suc 𝐴 ∈ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 6075 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | grusucd.1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Univ) | |
3 | grusucd.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐺) | |
4 | grusn 10075 | . . . 4 ⊢ ((𝐺 ∈ Univ ∧ 𝐴 ∈ 𝐺) → {𝐴} ∈ 𝐺) | |
5 | 2, 3, 4 | syl2anc 584 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝐺) |
6 | gruun 10077 | . . 3 ⊢ ((𝐺 ∈ Univ ∧ 𝐴 ∈ 𝐺 ∧ {𝐴} ∈ 𝐺) → (𝐴 ∪ {𝐴}) ∈ 𝐺) | |
7 | 2, 3, 5, 6 | syl3anc 1364 | . 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) ∈ 𝐺) |
8 | 1, 7 | syl5eqel 2886 | 1 ⊢ (𝜑 → suc 𝐴 ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2080 ∪ cun 3859 {csn 4474 suc csuc 6071 Univcgru 10061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-sbc 3708 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-fv 6236 df-ov 7022 df-oprab 7023 df-mpo 7024 df-map 8261 df-gru 10062 |
This theorem is referenced by: gruscottcld 40095 |
Copyright terms: Public domain | W3C validator |