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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 9391 | . . . . . 6 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ Inaccw) | |
2 | winaon 9389 | . . . . . 6 ⊢ (𝑥 ∈ Inaccw → 𝑥 ∈ On) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ On) |
4 | 3 | ssriv 3572 | . . . 4 ⊢ Inacc ⊆ On |
5 | ssorduni 6877 | . . . 4 ⊢ (Inacc ⊆ On → Ord ∪ Inacc) | |
6 | ordsson 6881 | . . . 4 ⊢ (Ord ∪ Inacc → ∪ Inacc ⊆ On) | |
7 | 4, 5, 6 | mp2b 10 | . . 3 ⊢ ∪ Inacc ⊆ On |
8 | vex 3176 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
9 | grothtsk 9536 | . . . . . . . 8 ⊢ ∪ Tarski = V | |
10 | 8, 9 | eleqtrri 2687 | . . . . . . 7 ⊢ 𝑦 ∈ ∪ Tarski |
11 | eluni2 4376 | . . . . . . 7 ⊢ (𝑦 ∈ ∪ Tarski ↔ ∃𝑤 ∈ Tarski 𝑦 ∈ 𝑤) | |
12 | 10, 11 | mpbi 219 | . . . . . 6 ⊢ ∃𝑤 ∈ Tarski 𝑦 ∈ 𝑤 |
13 | ne0i 3880 | . . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑤 → 𝑤 ≠ ∅) | |
14 | tskcard 9482 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ Tarski ∧ 𝑤 ≠ ∅) → (card‘𝑤) ∈ Inacc) | |
15 | 13, 14 | sylan2 490 | . . . . . . . . 9 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → (card‘𝑤) ∈ Inacc) |
16 | 15 | adantl 481 | . . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → (card‘𝑤) ∈ Inacc) |
17 | tsksdom 9457 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → 𝑦 ≺ 𝑤) | |
18 | 17 | adantl 481 | . . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → 𝑦 ≺ 𝑤) |
19 | tskwe2 9474 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ Tarski → 𝑤 ∈ dom card) | |
20 | 19 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤) → 𝑤 ∈ dom card) |
21 | cardsdomel 8683 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑤 ∈ dom card) → (𝑦 ≺ 𝑤 ↔ 𝑦 ∈ (card‘𝑤))) | |
22 | 20, 21 | sylan2 490 | . . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → (𝑦 ≺ 𝑤 ↔ 𝑦 ∈ (card‘𝑤))) |
23 | 18, 22 | mpbid 221 | . . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → 𝑦 ∈ (card‘𝑤)) |
24 | eleq2 2677 | . . . . . . . . 9 ⊢ (𝑧 = (card‘𝑤) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (card‘𝑤))) | |
25 | 24 | rspcev 3282 | . . . . . . . 8 ⊢ (((card‘𝑤) ∈ Inacc ∧ 𝑦 ∈ (card‘𝑤)) → ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) |
26 | 16, 23, 25 | syl2anc 691 | . . . . . . 7 ⊢ ((𝑦 ∈ On ∧ (𝑤 ∈ Tarski ∧ 𝑦 ∈ 𝑤)) → ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) |
27 | 26 | rexlimdvaa 3014 | . . . . . 6 ⊢ (𝑦 ∈ On → (∃𝑤 ∈ Tarski 𝑦 ∈ 𝑤 → ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧)) |
28 | 12, 27 | mpi 20 | . . . . 5 ⊢ (𝑦 ∈ On → ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) |
29 | eluni2 4376 | . . . . 5 ⊢ (𝑦 ∈ ∪ Inacc ↔ ∃𝑧 ∈ Inacc 𝑦 ∈ 𝑧) | |
30 | 28, 29 | sylibr 223 | . . . 4 ⊢ (𝑦 ∈ On → 𝑦 ∈ ∪ Inacc) |
31 | 30 | ssriv 3572 | . . 3 ⊢ On ⊆ ∪ Inacc |
32 | 7, 31 | eqssi 3584 | . 2 ⊢ ∪ Inacc = On |
33 | ssonprc 6884 | . . 3 ⊢ (Inacc ⊆ On → (Inacc ∉ V ↔ ∪ Inacc = On)) | |
34 | 4, 33 | ax-mp 5 | . 2 ⊢ (Inacc ∉ V ↔ ∪ Inacc = On) |
35 | 32, 34 | mpbir 220 | 1 ⊢ Inacc ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 class class class wbr 4583 dom cdm 5038 Ord word 5639 Oncon0 5640 ‘cfv 5804 ≺ csdm 7840 cardccrd 8644 Inaccwcwina 9383 Inacccina 9384 Tarskictsk 9449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-ac2 9168 ax-groth 9524 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-smo 7330 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-oi 8298 df-har 8346 df-r1 8510 df-card 8648 df-aleph 8649 df-cf 8650 df-acn 8651 df-ac 8822 df-wina 9385 df-ina 9386 df-tsk 9450 |
This theorem is referenced by: (None) |
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