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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 40649 | . 2 ⊢ (𝜑 → (𝑅1‘(rank‘𝐴)) ∈ 𝐺) |
4 | grurankrcld.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | r1rankid 9281 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) |
7 | gruss 10211 | . 2 ⊢ ((𝐺 ∈ Univ ∧ (𝑅1‘(rank‘𝐴)) ∈ 𝐺 ∧ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) → 𝐴 ∈ 𝐺) | |
8 | 1, 3, 6, 7 | syl3anc 1366 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ⊆ wss 3929 ‘cfv 6348 𝑅1cr1 9184 rankcrnk 9185 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-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-reg 9049 ax-inf2 9097 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-map 8401 df-r1 9186 df-rank 9187 df-gru 10206 |
This theorem is referenced by: gruscottcld 40666 |
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