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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gruscottcld | Structured version Visualization version GIF version |
Description: If a Grothendieck universe contains an element of a Scott's trick set, it contains the Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
gruscottcld.1 | ⊢ (𝜑 → 𝐺 ∈ Univ) |
gruscottcld.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐺) |
gruscottcld.3 | ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴) |
Ref | Expression |
---|---|
gruscottcld | ⊢ (𝜑 → Scott 𝐴 ∈ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gruscottcld.1 | . 2 ⊢ (𝜑 → 𝐺 ∈ Univ) | |
2 | gruscottcld.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴) | |
3 | 2 | scottrankd 44244 | . . 3 ⊢ (𝜑 → (rank‘Scott 𝐴) = suc (rank‘𝐵)) |
4 | gruscottcld.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐺) | |
5 | 1, 4 | grurankcld 44229 | . . . 4 ⊢ (𝜑 → (rank‘𝐵) ∈ 𝐺) |
6 | 1, 5 | grusucd 44226 | . . 3 ⊢ (𝜑 → suc (rank‘𝐵) ∈ 𝐺) |
7 | 3, 6 | eqeltrd 2839 | . 2 ⊢ (𝜑 → (rank‘Scott 𝐴) ∈ 𝐺) |
8 | scottex2 44241 | . . 3 ⊢ Scott 𝐴 ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → Scott 𝐴 ∈ V) |
10 | 1, 7, 9 | grurankrcld 44230 | 1 ⊢ (𝜑 → Scott 𝐴 ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3478 suc csuc 6388 ‘cfv 6563 rankcrnk 9801 Univcgru 10828 Scott cscott 44231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-reg 9630 ax-inf2 9679 ax-ac2 10501 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-tc 9775 df-r1 9802 df-rank 9803 df-card 9977 df-cf 9979 df-acn 9980 df-ac 10154 df-wina 10722 df-ina 10723 df-gru 10829 df-scott 44232 |
This theorem is referenced by: grucollcld 44256 |
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