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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 10626 | . . . . . 6 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ Inaccw) | |
2 | winaon 10624 | . . . . . 6 ⊢ (𝑥 ∈ Inaccw → 𝑥 ∈ On) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ On) |
4 | 3 | ssriv 3948 | . . . 4 ⊢ Inacc ⊆ On |
5 | ssorduni 7713 | . . . 4 ⊢ (Inacc ⊆ On → Ord ∪ Inacc) | |
6 | ordsson 7717 | . . . 4 ⊢ (Ord ∪ Inacc → ∪ Inacc ⊆ On) | |
7 | 4, 5, 6 | mp2b 10 | . . 3 ⊢ ∪ Inacc ⊆ On |
8 | vex 3449 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
9 | grothtsk 10771 | . . . . . . . 8 ⊢ ∪ Tarski = V | |
10 | 8, 9 | eleqtrri 2837 | . . . . . . 7 ⊢ 𝑦 ∈ ∪ Tarski |
11 | eluni2 4869 | . . . . . . 7 ⊢ (𝑦 ∈ ∪ Tarski ↔ ∃𝑤 ∈ Tarski 𝑦 ∈ 𝑤) | |
12 | 10, 11 | mpbi 229 | . . . . . 6 ⊢ ∃𝑤 ∈ Tarski 𝑦 ∈ 𝑤 |
13 | ne0i 4294 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝑤 → 𝑤 ≠ ∅) | |
14 | tskcard 10717 | . . . . . . . . 9 ⊢ ((𝑤 ∈ Tarski ∧ 𝑤 ≠ ∅) → (card‘𝑤) ∈ Inacc) | |
15 | 13, 14 | sylan2 593 | . . . . . . . 8 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → (card‘𝑤) ∈ Inacc) |
16 | tsksdom 10692 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → 𝑦 ≺ 𝑤) | |
17 | 16 | adantl 482 | . . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → 𝑦 ≺ 𝑤) |
18 | tskwe2 10709 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ Tarski → 𝑤 ∈ dom card) | |
19 | 18 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → 𝑤 ∈ dom card) |
20 | cardsdomel 9910 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑤 ∈ dom card) → (𝑦 ≺ 𝑤 ↔ 𝑦 ∈ (card‘𝑤))) | |
21 | 19, 20 | sylan2 593 | . . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → (𝑦 ≺ 𝑤 ↔ 𝑦 ∈ (card‘𝑤))) |
22 | 17, 21 | mpbid 231 | . . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → 𝑦 ∈ (card‘𝑤)) |
23 | eleq2 2826 | . . . . . . . . 9 ⊢ (𝑧 = (card‘𝑤) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (card‘𝑤))) | |
24 | 23 | rspcev 3581 | . . . . . . . 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 3153 | . . . . . 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 4869 | . . . . 5 ⊢ (𝑦 ∈ ∪ Inacc ↔ ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) | |
29 | 27, 28 | sylibr 233 | . . . 4 ⊢ (𝑦 ∈ On → 𝑦 ∈ ∪ Inacc) |
30 | 29 | ssriv 3948 | . . 3 ⊢ On ⊆ ∪ Inacc |
31 | 7, 30 | eqssi 3960 | . 2 ⊢ ∪ Inacc = On |
32 | ssonprc 7722 | . . 3 ⊢ (Inacc ⊆ On → (Inacc ∉ V ↔ ∪ Inacc = On)) | |
33 | 4, 32 | ax-mp 5 | . 2 ⊢ (Inacc ∉ V ↔ ∪ Inacc = On) |
34 | 31, 33 | mpbir 230 | 1 ⊢ Inacc ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∉ wnel 3049 ∃wrex 3073 Vcvv 3445 ⊆ wss 3910 ∅c0 4282 ∪ cuni 4865 class class class wbr 5105 dom cdm 5633 Ord word 6316 Oncon0 6317 ‘cfv 6496 ≺ csdm 8882 cardccrd 9871 Inaccwcwina 10618 Inacccina 10619 Tarskictsk 10684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-ac2 10399 ax-groth 10759 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-smo 8292 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-oi 9446 df-har 9493 df-r1 9700 df-card 9875 df-aleph 9876 df-cf 9877 df-acn 9878 df-ac 10052 df-wina 10620 df-ina 10621 df-tsk 10685 |
This theorem is referenced by: inaex 42567 |
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