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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 6377 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | grusucd.1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Univ) | |
3 | grusucd.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐺) | |
4 | grusn 10829 | . . . 4 ⊢ ((𝐺 ∈ Univ ∧ 𝐴 ∈ 𝐺) → {𝐴} ∈ 𝐺) | |
5 | 2, 3, 4 | syl2anc 582 | . . 3 ⊢ (𝜑 → {𝐴} ∈ 𝐺) |
6 | gruun 10831 | . . 3 ⊢ ((𝐺 ∈ Univ ∧ 𝐴 ∈ 𝐺 ∧ {𝐴} ∈ 𝐺) → (𝐴 ∪ {𝐴}) ∈ 𝐺) | |
7 | 2, 3, 5, 6 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) ∈ 𝐺) |
8 | 1, 7 | eqeltrid 2829 | 1 ⊢ (𝜑 → suc 𝐴 ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∪ cun 3942 {csn 4630 suc csuc 6373 Univcgru 10815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-map 8847 df-gru 10816 |
This theorem is referenced by: gruscottcld 43828 |
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