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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 10728 | . . . . . 6 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ Inaccw) | |
2 | winaon 10726 | . . . . . 6 ⊢ (𝑥 ∈ Inaccw → 𝑥 ∈ On) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ On) |
4 | 3 | ssriv 3999 | . . . 4 ⊢ Inacc ⊆ On |
5 | ssorduni 7798 | . . . 4 ⊢ (Inacc ⊆ On → Ord ∪ Inacc) | |
6 | ordsson 7802 | . . . 4 ⊢ (Ord ∪ Inacc → ∪ Inacc ⊆ On) | |
7 | 4, 5, 6 | mp2b 10 | . . 3 ⊢ ∪ Inacc ⊆ On |
8 | vex 3482 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
9 | grothtsk 10873 | . . . . . . . 8 ⊢ ∪ Tarski = V | |
10 | 8, 9 | eleqtrri 2838 | . . . . . . 7 ⊢ 𝑦 ∈ ∪ Tarski |
11 | eluni2 4916 | . . . . . . 7 ⊢ (𝑦 ∈ ∪ Tarski ↔ ∃𝑤 ∈ Tarski 𝑦 ∈ 𝑤) | |
12 | 10, 11 | mpbi 230 | . . . . . 6 ⊢ ∃𝑤 ∈ Tarski 𝑦 ∈ 𝑤 |
13 | ne0i 4347 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝑤 → 𝑤 ≠ ∅) | |
14 | tskcard 10819 | . . . . . . . . 9 ⊢ ((𝑤 ∈ Tarski ∧ 𝑤 ≠ ∅) → (card‘𝑤) ∈ Inacc) | |
15 | 13, 14 | sylan2 593 | . . . . . . . 8 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → (card‘𝑤) ∈ Inacc) |
16 | tsksdom 10794 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → 𝑦 ≺ 𝑤) | |
17 | 16 | adantl 481 | . . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → 𝑦 ≺ 𝑤) |
18 | tskwe2 10811 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ Tarski → 𝑤 ∈ dom card) | |
19 | 18 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → 𝑤 ∈ dom card) |
20 | cardsdomel 10012 | . . . . . . . . . 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 232 | . . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → 𝑦 ∈ (card‘𝑤)) |
23 | eleq2 2828 | . . . . . . . . 9 ⊢ (𝑧 = (card‘𝑤) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (card‘𝑤))) | |
24 | 23 | rspcev 3622 | . . . . . . . 8 ⊢ (((card‘𝑤) ∈ Inacc ∧ 𝑦 ∈ (card‘𝑤)) → ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) |
25 | 15, 22, 24 | syl2an2 686 | . . . . . . 7 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) |
26 | 25 | rexlimdvaa 3154 | . . . . . 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 4916 | . . . . 5 ⊢ (𝑦 ∈ ∪ Inacc ↔ ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) | |
29 | 27, 28 | sylibr 234 | . . . 4 ⊢ (𝑦 ∈ On → 𝑦 ∈ ∪ Inacc) |
30 | 29 | ssriv 3999 | . . 3 ⊢ On ⊆ ∪ Inacc |
31 | 7, 30 | eqssi 4012 | . 2 ⊢ ∪ Inacc = On |
32 | ssonprc 7807 | . . 3 ⊢ (Inacc ⊆ On → (Inacc ∉ V ↔ ∪ Inacc = On)) | |
33 | 4, 32 | ax-mp 5 | . 2 ⊢ (Inacc ∉ V ↔ ∪ Inacc = On) |
34 | 31, 33 | mpbir 231 | 1 ⊢ Inacc ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∉ wnel 3044 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 ∪ cuni 4912 class class class wbr 5148 dom cdm 5689 Ord word 6385 Oncon0 6386 ‘cfv 6563 ≺ csdm 8983 cardccrd 9973 Inaccwcwina 10720 Inacccina 10721 Tarskictsk 10786 |
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-inf2 9679 ax-ac2 10501 ax-groth 10861 |
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-nel 3045 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-smo 8385 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-oi 9548 df-har 9595 df-r1 9802 df-card 9977 df-aleph 9978 df-cf 9979 df-acn 9980 df-ac 10154 df-wina 10722 df-ina 10723 df-tsk 10787 |
This theorem is referenced by: inaex 44293 |
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