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Mirrors > Home > MPE Home > Th. List > Mathboxes > grurankcld | Structured version Visualization version GIF version |
Description: Grothendieck universes are closed under the rank function. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
Ref | Expression |
---|---|
grurankcld.1 | ⊢ (𝜑 → 𝐺 ∈ Univ) |
grurankcld.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐺) |
Ref | Expression |
---|---|
grurankcld | ⊢ (𝜑 → (rank‘𝐴) ∈ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grurankcld.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐺) | |
2 | grurankcld.1 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Univ) | |
3 | 2 | elexd 3511 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V) |
4 | unir1 9235 | . . . . . 6 ⊢ ∪ (𝑅1 “ On) = V | |
5 | 3, 4 | eleqtrrdi 2923 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ ∪ (𝑅1 “ On)) |
6 | eqid 2820 | . . . . . 6 ⊢ (𝐺 ∩ On) = (𝐺 ∩ On) | |
7 | 6 | grur1 10235 | . . . . 5 ⊢ ((𝐺 ∈ Univ ∧ 𝐺 ∈ ∪ (𝑅1 “ On)) → 𝐺 = (𝑅1‘(𝐺 ∩ On))) |
8 | 2, 5, 7 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐺 = (𝑅1‘(𝐺 ∩ On))) |
9 | 1, 8 | eleqtrd 2914 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑅1‘(𝐺 ∩ On))) |
10 | 9 | r1rankcld 40641 | . 2 ⊢ (𝜑 → (rank‘𝐴) ∈ (𝑅1‘(𝐺 ∩ On))) |
11 | 10, 8 | eleqtrrd 2915 | 1 ⊢ (𝜑 → (rank‘𝐴) ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∩ cin 3928 ∪ cuni 4831 “ cima 5551 Oncon0 6184 ‘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 ax-ac2 9878 |
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-rmo 3145 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-iin 4915 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-se 5508 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-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-tc 9172 df-r1 9186 df-rank 9187 df-card 9361 df-cf 9363 df-acn 9364 df-ac 9535 df-wina 10099 df-ina 10100 df-gru 10206 |
This theorem is referenced by: gruscottcld 40659 |
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