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| Mirrors > Home > MPE Home > Th. List > Mathboxes > grurankrcld | Structured version Visualization version GIF version | ||
| Description: If a Grothendieck universe contains a set's rank, it contains that set. (Contributed by Rohan Ridenour, 9-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| grurankrcld.1 | ⊢ (𝜑 → 𝐺 ∈ Univ) | 
| grurankrcld.2 | ⊢ (𝜑 → (rank‘𝐴) ∈ 𝐺) | 
| grurankrcld.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| Ref | Expression | 
|---|---|
| grurankrcld | ⊢ (𝜑 → 𝐴 ∈ 𝐺) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | grurankrcld.1 | . 2 ⊢ (𝜑 → 𝐺 ∈ Univ) | |
| 2 | grurankrcld.2 | . . 3 ⊢ (𝜑 → (rank‘𝐴) ∈ 𝐺) | |
| 3 | 1, 2 | grur1cld 44256 | . 2 ⊢ (𝜑 → (𝑅1‘(rank‘𝐴)) ∈ 𝐺) | 
| 4 | grurankrcld.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 5 | r1rankid 9900 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
| 6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | 
| 7 | gruss 10837 | . 2 ⊢ ((𝐺 ∈ Univ ∧ (𝑅1‘(rank‘𝐴)) ∈ 𝐺 ∧ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) → 𝐴 ∈ 𝐺) | |
| 8 | 1, 3, 6, 7 | syl3anc 1372 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐺) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ⊆ wss 3950 ‘cfv 6560 𝑅1cr1 9803 rankcrnk 9804 Univcgru 10831 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-reg 9633 ax-inf2 9682 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-map 8869 df-r1 9805 df-rank 9806 df-gru 10832 | 
| This theorem is referenced by: gruscottcld 44273 | 
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