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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 10730 | . . . . . 6 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ Inaccw) | |
| 2 | winaon 10728 | . . . . . 6 ⊢ (𝑥 ∈ Inaccw → 𝑥 ∈ On) | |
| 3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ On) |
| 4 | 3 | ssriv 3987 | . . . 4 ⊢ Inacc ⊆ On |
| 5 | ssorduni 7799 | . . . 4 ⊢ (Inacc ⊆ On → Ord ∪ Inacc) | |
| 6 | ordsson 7803 | . . . 4 ⊢ (Ord ∪ Inacc → ∪ Inacc ⊆ On) | |
| 7 | 4, 5, 6 | mp2b 10 | . . 3 ⊢ ∪ Inacc ⊆ On |
| 8 | vex 3484 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
| 9 | grothtsk 10875 | . . . . . . . 8 ⊢ ∪ Tarski = V | |
| 10 | 8, 9 | eleqtrri 2840 | . . . . . . 7 ⊢ 𝑦 ∈ ∪ Tarski |
| 11 | eluni2 4911 | . . . . . . 7 ⊢ (𝑦 ∈ ∪ Tarski ↔ ∃𝑤 ∈ Tarski 𝑦 ∈ 𝑤) | |
| 12 | 10, 11 | mpbi 230 | . . . . . 6 ⊢ ∃𝑤 ∈ Tarski 𝑦 ∈ 𝑤 |
| 13 | ne0i 4341 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝑤 → 𝑤 ≠ ∅) | |
| 14 | tskcard 10821 | . . . . . . . . 9 ⊢ ((𝑤 ∈ Tarski ∧ 𝑤 ≠ ∅) → (card‘𝑤) ∈ Inacc) | |
| 15 | 13, 14 | sylan2 593 | . . . . . . . 8 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → (card‘𝑤) ∈ Inacc) |
| 16 | tsksdom 10796 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → 𝑦 ≺ 𝑤) | |
| 17 | 16 | adantl 481 | . . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → 𝑦 ≺ 𝑤) |
| 18 | tskwe2 10813 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ Tarski → 𝑤 ∈ dom card) | |
| 19 | 18 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → 𝑤 ∈ dom card) |
| 20 | cardsdomel 10014 | . . . . . . . . . 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 2830 | . . . . . . . . 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 3156 | . . . . . 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 4911 | . . . . 5 ⊢ (𝑦 ∈ ∪ Inacc ↔ ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) | |
| 29 | 27, 28 | sylibr 234 | . . . 4 ⊢ (𝑦 ∈ On → 𝑦 ∈ ∪ Inacc) |
| 30 | 29 | ssriv 3987 | . . 3 ⊢ On ⊆ ∪ Inacc |
| 31 | 7, 30 | eqssi 4000 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∉ wnel 3046 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 class class class wbr 5143 dom cdm 5685 Ord word 6383 Oncon0 6384 ‘cfv 6561 ≺ csdm 8984 cardccrd 9975 Inaccwcwina 10722 Inacccina 10723 Tarskictsk 10788 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-ac2 10503 ax-groth 10863 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-smo 8386 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-oi 9550 df-har 9597 df-r1 9804 df-card 9979 df-aleph 9980 df-cf 9981 df-acn 9982 df-ac 10156 df-wina 10724 df-ina 10725 df-tsk 10789 |
| This theorem is referenced by: inaex 44316 |
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