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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 10626 | . . . . . . 7 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ Inaccw) | |
2 | winaon 10624 | . . . . . . 7 ⊢ (𝑥 ∈ Inaccw → 𝑥 ∈ On) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ On) |
4 | 3 | ssriv 3948 | . . . . 5 ⊢ Inacc ⊆ On |
5 | onmindif 6409 | . . . . 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 2841 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝑥 = ∩ (Inacc ∖ suc 𝐴)) → 𝐴 ∈ 𝑥) |
10 | difss 4091 | . . . . 5 ⊢ (Inacc ∖ suc 𝐴) ⊆ Inacc | |
11 | 10, 4 | sstri 3953 | . . . 4 ⊢ (Inacc ∖ suc 𝐴) ⊆ On |
12 | inaprc 10772 | . . . . . . 7 ⊢ Inacc ∉ V | |
13 | 12 | neli 3051 | . . . . . 6 ⊢ ¬ Inacc ∈ V |
14 | ssdif0 4323 | . . . . . . 7 ⊢ (Inacc ⊆ suc 𝐴 ↔ (Inacc ∖ suc 𝐴) = ∅) | |
15 | sucexg 7740 | . . . . . . . 8 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V) | |
16 | ssexg 5280 | . . . . . . . . 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 2950 | . . . 4 ⊢ (𝐴 ∈ On → (Inacc ∖ suc 𝐴) ≠ ∅) |
22 | onint 7725 | . . . 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 3922 | . 2 ⊢ (𝐴 ∈ On → ∩ (Inacc ∖ suc 𝐴) ∈ Inacc) |
25 | 9, 24 | rspcime 3584 | 1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ Inacc 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 Vcvv 3445 ∖ cdif 3907 ⊆ wss 3910 ∅c0 4282 ∩ cint 4907 Oncon0 6317 suc csuc 6319 Inaccwcwina 10618 Inacccina 10619 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-ac2 10399 ax-groth 10759 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-smo 8292 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-oi 9446 df-har 9493 df-r1 9700 df-card 9875 df-aleph 9876 df-cf 9877 df-acn 9878 df-ac 10052 df-wina 10620 df-ina 10621 df-tsk 10685 |
This theorem is referenced by: gruex 42568 |
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