Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > inaprc | Structured version Visualization version GIF version |
Description: An equivalent to the Tarski-Grothendieck Axiom: there is a proper class of inaccessible cardinals. (Contributed by Mario Carneiro, 9-Jun-2013.) |
Ref | Expression |
---|---|
inaprc | ⊢ Inacc ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inawina 10114 | . . . . . 6 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ Inaccw) | |
2 | winaon 10112 | . . . . . 6 ⊢ (𝑥 ∈ Inaccw → 𝑥 ∈ On) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ On) |
4 | 3 | ssriv 3973 | . . . 4 ⊢ Inacc ⊆ On |
5 | ssorduni 7502 | . . . 4 ⊢ (Inacc ⊆ On → Ord ∪ Inacc) | |
6 | ordsson 7506 | . . . 4 ⊢ (Ord ∪ Inacc → ∪ Inacc ⊆ On) | |
7 | 4, 5, 6 | mp2b 10 | . . 3 ⊢ ∪ Inacc ⊆ On |
8 | vex 3499 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
9 | grothtsk 10259 | . . . . . . . 8 ⊢ ∪ Tarski = V | |
10 | 8, 9 | eleqtrri 2914 | . . . . . . 7 ⊢ 𝑦 ∈ ∪ Tarski |
11 | eluni2 4844 | . . . . . . 7 ⊢ (𝑦 ∈ ∪ Tarski ↔ ∃𝑤 ∈ Tarski 𝑦 ∈ 𝑤) | |
12 | 10, 11 | mpbi 232 | . . . . . 6 ⊢ ∃𝑤 ∈ Tarski 𝑦 ∈ 𝑤 |
13 | ne0i 4302 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝑤 → 𝑤 ≠ ∅) | |
14 | tskcard 10205 | . . . . . . . . 9 ⊢ ((𝑤 ∈ Tarski ∧ 𝑤 ≠ ∅) → (card‘𝑤) ∈ Inacc) | |
15 | 13, 14 | sylan2 594 | . . . . . . . 8 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → (card‘𝑤) ∈ Inacc) |
16 | tsksdom 10180 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → 𝑦 ≺ 𝑤) | |
17 | 16 | adantl 484 | . . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → 𝑦 ≺ 𝑤) |
18 | tskwe2 10197 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ Tarski → 𝑤 ∈ dom card) | |
19 | 18 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → 𝑤 ∈ dom card) |
20 | cardsdomel 9405 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑤 ∈ dom card) → (𝑦 ≺ 𝑤 ↔ 𝑦 ∈ (card‘𝑤))) | |
21 | 19, 20 | sylan2 594 | . . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → (𝑦 ≺ 𝑤 ↔ 𝑦 ∈ (card‘𝑤))) |
22 | 17, 21 | mpbid 234 | . . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → 𝑦 ∈ (card‘𝑤)) |
23 | eleq2 2903 | . . . . . . . . 9 ⊢ (𝑧 = (card‘𝑤) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (card‘𝑤))) | |
24 | 23 | rspcev 3625 | . . . . . . . 8 ⊢ (((card‘𝑤) ∈ Inacc ∧ 𝑦 ∈ (card‘𝑤)) → ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) |
25 | 15, 22, 24 | syl2an2 684 | . . . . . . 7 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) |
26 | 25 | rexlimdvaa 3287 | . . . . . 6 ⊢ (𝑦 ∈ On → (∃𝑤 ∈ Tarski 𝑦 ∈ 𝑤 → ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧)) |
27 | 12, 26 | mpi 20 | . . . . 5 ⊢ (𝑦 ∈ On → ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) |
28 | eluni2 4844 | . . . . 5 ⊢ (𝑦 ∈ ∪ Inacc ↔ ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) | |
29 | 27, 28 | sylibr 236 | . . . 4 ⊢ (𝑦 ∈ On → 𝑦 ∈ ∪ Inacc) |
30 | 29 | ssriv 3973 | . . 3 ⊢ On ⊆ ∪ Inacc |
31 | 7, 30 | eqssi 3985 | . 2 ⊢ ∪ Inacc = On |
32 | ssonprc 7509 | . . 3 ⊢ (Inacc ⊆ On → (Inacc ∉ V ↔ ∪ Inacc = On)) | |
33 | 4, 32 | ax-mp 5 | . 2 ⊢ (Inacc ∉ V ↔ ∪ Inacc = On) |
34 | 31, 33 | mpbir 233 | 1 ⊢ Inacc ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∉ wnel 3125 ∃wrex 3141 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 ∪ cuni 4840 class class class wbr 5068 dom cdm 5557 Ord word 6192 Oncon0 6193 ‘cfv 6357 ≺ csdm 8510 cardccrd 9366 Inaccwcwina 10106 Inacccina 10107 Tarskictsk 10172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-ac2 9887 ax-groth 10247 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-smo 7985 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-har 9024 df-r1 9195 df-card 9370 df-aleph 9371 df-cf 9372 df-acn 9373 df-ac 9544 df-wina 10108 df-ina 10109 df-tsk 10173 |
This theorem is referenced by: inaex 40640 |
Copyright terms: Public domain | W3C validator |