![]() |
Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > inaex | Structured version Visualization version GIF version |
Description: Assuming the Tarski-Grothendieck axiom, every ordinal is contained in an inaccessible ordinal. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
inaex | ⊢ (𝐴 ∈ On → ∃𝑥 ∈ Inacc 𝐴 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inawina 10727 | . . . . . . 7 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ Inaccw) | |
2 | winaon 10725 | . . . . . . 7 ⊢ (𝑥 ∈ Inaccw → 𝑥 ∈ On) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ On) |
4 | 3 | ssriv 3998 | . . . . 5 ⊢ Inacc ⊆ On |
5 | onmindif 6477 | . . . . 5 ⊢ ((Inacc ⊆ On ∧ 𝐴 ∈ On) → 𝐴 ∈ ∩ (Inacc ∖ suc 𝐴)) | |
6 | 4, 5 | mpan 690 | . . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ ∩ (Inacc ∖ suc 𝐴)) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 = ∩ (Inacc ∖ suc 𝐴)) → 𝐴 ∈ ∩ (Inacc ∖ suc 𝐴)) |
8 | simpr 484 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 = ∩ (Inacc ∖ suc 𝐴)) → 𝑥 = ∩ (Inacc ∖ suc 𝐴)) | |
9 | 7, 8 | eleqtrrd 2841 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝑥 = ∩ (Inacc ∖ suc 𝐴)) → 𝐴 ∈ 𝑥) |
10 | difss 4145 | . . . . 5 ⊢ (Inacc ∖ suc 𝐴) ⊆ Inacc | |
11 | 10, 4 | sstri 4004 | . . . 4 ⊢ (Inacc ∖ suc 𝐴) ⊆ On |
12 | inaprc 10873 | . . . . . . 7 ⊢ Inacc ∉ V | |
13 | 12 | neli 3045 | . . . . . 6 ⊢ ¬ Inacc ∈ V |
14 | ssdif0 4371 | . . . . . . 7 ⊢ (Inacc ⊆ suc 𝐴 ↔ (Inacc ∖ suc 𝐴) = ∅) | |
15 | sucexg 7824 | . . . . . . . 8 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V) | |
16 | ssexg 5328 | . . . . . . . . 9 ⊢ ((Inacc ⊆ suc 𝐴 ∧ suc 𝐴 ∈ V) → Inacc ∈ V) | |
17 | 16 | expcom 413 | . . . . . . . 8 ⊢ (suc 𝐴 ∈ V → (Inacc ⊆ suc 𝐴 → Inacc ∈ V)) |
18 | 15, 17 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ On → (Inacc ⊆ suc 𝐴 → Inacc ∈ V)) |
19 | 14, 18 | biimtrrid 243 | . . . . . 6 ⊢ (𝐴 ∈ On → ((Inacc ∖ suc 𝐴) = ∅ → Inacc ∈ V)) |
20 | 13, 19 | mtoi 199 | . . . . 5 ⊢ (𝐴 ∈ On → ¬ (Inacc ∖ suc 𝐴) = ∅) |
21 | 20 | neqned 2944 | . . . 4 ⊢ (𝐴 ∈ On → (Inacc ∖ suc 𝐴) ≠ ∅) |
22 | onint 7809 | . . . 4 ⊢ (((Inacc ∖ suc 𝐴) ⊆ On ∧ (Inacc ∖ suc 𝐴) ≠ ∅) → ∩ (Inacc ∖ suc 𝐴) ∈ (Inacc ∖ suc 𝐴)) | |
23 | 11, 21, 22 | sylancr 587 | . . 3 ⊢ (𝐴 ∈ On → ∩ (Inacc ∖ suc 𝐴) ∈ (Inacc ∖ suc 𝐴)) |
24 | 23 | eldifad 3974 | . 2 ⊢ (𝐴 ∈ On → ∩ (Inacc ∖ suc 𝐴) ∈ Inacc) |
25 | 9, 24 | rspcime 3626 | 1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ Inacc 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 Vcvv 3477 ∖ cdif 3959 ⊆ wss 3962 ∅c0 4338 ∩ cint 4950 Oncon0 6385 suc csuc 6387 Inaccwcwina 10719 Inacccina 10720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-ac2 10500 ax-groth 10860 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-smo 8384 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-oi 9547 df-har 9594 df-r1 9801 df-card 9976 df-aleph 9977 df-cf 9978 df-acn 9979 df-ac 10153 df-wina 10721 df-ina 10722 df-tsk 10786 |
This theorem is referenced by: gruex 44293 |
Copyright terms: Public domain | W3C validator |