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Mathbox for Rohan Ridenour |
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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 10684 | . . . . . . 7 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ Inaccw) | |
2 | winaon 10682 | . . . . . . 7 ⊢ (𝑥 ∈ Inaccw → 𝑥 ∈ On) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ On) |
4 | 3 | ssriv 3986 | . . . . 5 ⊢ Inacc ⊆ On |
5 | onmindif 6456 | . . . . 5 ⊢ ((Inacc ⊆ On ∧ 𝐴 ∈ On) → 𝐴 ∈ ∩ (Inacc ∖ suc 𝐴)) | |
6 | 4, 5 | mpan 688 | . . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ ∩ (Inacc ∖ suc 𝐴)) |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 = ∩ (Inacc ∖ suc 𝐴)) → 𝐴 ∈ ∩ (Inacc ∖ suc 𝐴)) |
8 | simpr 485 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 = ∩ (Inacc ∖ suc 𝐴)) → 𝑥 = ∩ (Inacc ∖ suc 𝐴)) | |
9 | 7, 8 | eleqtrrd 2836 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝑥 = ∩ (Inacc ∖ suc 𝐴)) → 𝐴 ∈ 𝑥) |
10 | difss 4131 | . . . . 5 ⊢ (Inacc ∖ suc 𝐴) ⊆ Inacc | |
11 | 10, 4 | sstri 3991 | . . . 4 ⊢ (Inacc ∖ suc 𝐴) ⊆ On |
12 | inaprc 10830 | . . . . . . 7 ⊢ Inacc ∉ V | |
13 | 12 | neli 3048 | . . . . . 6 ⊢ ¬ Inacc ∈ V |
14 | ssdif0 4363 | . . . . . . 7 ⊢ (Inacc ⊆ suc 𝐴 ↔ (Inacc ∖ suc 𝐴) = ∅) | |
15 | sucexg 7792 | . . . . . . . 8 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V) | |
16 | ssexg 5323 | . . . . . . . . 9 ⊢ ((Inacc ⊆ suc 𝐴 ∧ suc 𝐴 ∈ V) → Inacc ∈ V) | |
17 | 16 | expcom 414 | . . . . . . . 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 242 | . . . . . 6 ⊢ (𝐴 ∈ On → ((Inacc ∖ suc 𝐴) = ∅ → Inacc ∈ V)) |
20 | 13, 19 | mtoi 198 | . . . . 5 ⊢ (𝐴 ∈ On → ¬ (Inacc ∖ suc 𝐴) = ∅) |
21 | 20 | neqned 2947 | . . . 4 ⊢ (𝐴 ∈ On → (Inacc ∖ suc 𝐴) ≠ ∅) |
22 | onint 7777 | . . . 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 3960 | . 2 ⊢ (𝐴 ∈ On → ∩ (Inacc ∖ suc 𝐴) ∈ Inacc) |
25 | 9, 24 | rspcime 3616 | 1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ Inacc 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 Vcvv 3474 ∖ cdif 3945 ⊆ wss 3948 ∅c0 4322 ∩ cint 4950 Oncon0 6364 suc csuc 6366 Inaccwcwina 10676 Inacccina 10677 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-ac2 10457 ax-groth 10817 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-smo 8345 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-oi 9504 df-har 9551 df-r1 9758 df-card 9933 df-aleph 9934 df-cf 9935 df-acn 9936 df-ac 10110 df-wina 10678 df-ina 10679 df-tsk 10743 |
This theorem is referenced by: gruex 43047 |
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