Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gruex | Structured version Visualization version GIF version |
Description: Assuming the Tarski-Grothendieck axiom, every set is contained in a Grothendieck universe. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
gruex | ⊢ ∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankon 9217 | . . 3 ⊢ (rank‘𝑥) ∈ On | |
2 | inaex 40708 | . . 3 ⊢ ((rank‘𝑥) ∈ On → ∃𝑧 ∈ Inacc (rank‘𝑥) ∈ 𝑧) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∃𝑧 ∈ Inacc (rank‘𝑥) ∈ 𝑧 |
4 | simplr 767 | . . . . . 6 ⊢ (((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) ∧ 𝑦 = (𝑅1‘𝑧)) → (rank‘𝑥) ∈ 𝑧) | |
5 | inawina 10105 | . . . . . . . . 9 ⊢ (𝑧 ∈ Inacc → 𝑧 ∈ Inaccw) | |
6 | winaon 10103 | . . . . . . . . 9 ⊢ (𝑧 ∈ Inaccw → 𝑧 ∈ On) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (𝑧 ∈ Inacc → 𝑧 ∈ On) |
8 | 7 | ad2antrr 724 | . . . . . . 7 ⊢ (((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) ∧ 𝑦 = (𝑅1‘𝑧)) → 𝑧 ∈ On) |
9 | vex 3494 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
10 | 9 | rankr1a 9258 | . . . . . . 7 ⊢ (𝑧 ∈ On → (𝑥 ∈ (𝑅1‘𝑧) ↔ (rank‘𝑥) ∈ 𝑧)) |
11 | 8, 10 | syl 17 | . . . . . 6 ⊢ (((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) ∧ 𝑦 = (𝑅1‘𝑧)) → (𝑥 ∈ (𝑅1‘𝑧) ↔ (rank‘𝑥) ∈ 𝑧)) |
12 | 4, 11 | mpbird 259 | . . . . 5 ⊢ (((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) ∧ 𝑦 = (𝑅1‘𝑧)) → 𝑥 ∈ (𝑅1‘𝑧)) |
13 | simpr 487 | . . . . 5 ⊢ (((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) ∧ 𝑦 = (𝑅1‘𝑧)) → 𝑦 = (𝑅1‘𝑧)) | |
14 | 12, 13 | eleqtrrd 2915 | . . . 4 ⊢ (((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) ∧ 𝑦 = (𝑅1‘𝑧)) → 𝑥 ∈ 𝑦) |
15 | simpl 485 | . . . . 5 ⊢ ((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) → 𝑧 ∈ Inacc) | |
16 | 15 | inagrud 40707 | . . . 4 ⊢ ((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) → (𝑅1‘𝑧) ∈ Univ) |
17 | 14, 16 | rspcime 3624 | . . 3 ⊢ ((𝑧 ∈ Inacc ∧ (rank‘𝑥) ∈ 𝑧) → ∃𝑦 ∈ Univ 𝑥 ∈ 𝑦) |
18 | 17 | rexlimiva 3280 | . 2 ⊢ (∃𝑧 ∈ Inacc (rank‘𝑥) ∈ 𝑧 → ∃𝑦 ∈ Univ 𝑥 ∈ 𝑦) |
19 | 3, 18 | ax-mp 5 | 1 ⊢ ∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 Oncon0 6184 ‘cfv 6348 𝑅1cr1 9184 rankcrnk 9185 Inaccwcwina 10097 Inacccina 10098 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 ax-groth 10238 |
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-nel 3123 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-smo 7976 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-oi 8967 df-har 9015 df-r1 9186 df-rank 9187 df-card 9361 df-aleph 9362 df-cf 9363 df-acn 9364 df-ac 9535 df-wina 10099 df-ina 10100 df-tsk 10164 df-gru 10206 |
This theorem is referenced by: rr-groth 40710 rr-grothprim 40711 |
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