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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 10730 | . . . . . . 7 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ Inaccw) | |
| 2 | winaon 10728 | . . . . . . 7 ⊢ (𝑥 ∈ Inaccw → 𝑥 ∈ On) | |
| 3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ Inacc → 𝑥 ∈ On) |
| 4 | 3 | ssriv 3987 | . . . . 5 ⊢ Inacc ⊆ On |
| 5 | onmindif 6476 | . . . . 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 2844 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝑥 = ∩ (Inacc ∖ suc 𝐴)) → 𝐴 ∈ 𝑥) |
| 10 | difss 4136 | . . . . 5 ⊢ (Inacc ∖ suc 𝐴) ⊆ Inacc | |
| 11 | 10, 4 | sstri 3993 | . . . 4 ⊢ (Inacc ∖ suc 𝐴) ⊆ On |
| 12 | inaprc 10876 | . . . . . . 7 ⊢ Inacc ∉ V | |
| 13 | 12 | neli 3048 | . . . . . 6 ⊢ ¬ Inacc ∈ V |
| 14 | ssdif0 4366 | . . . . . . 7 ⊢ (Inacc ⊆ suc 𝐴 ↔ (Inacc ∖ suc 𝐴) = ∅) | |
| 15 | sucexg 7825 | . . . . . . . 8 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V) | |
| 16 | ssexg 5323 | . . . . . . . . 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 2947 | . . . 4 ⊢ (𝐴 ∈ On → (Inacc ∖ suc 𝐴) ≠ ∅) |
| 22 | onint 7810 | . . . 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 3963 | . 2 ⊢ (𝐴 ∈ On → ∩ (Inacc ∖ suc 𝐴) ∈ Inacc) |
| 25 | 9, 24 | rspcime 3627 | 1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ Inacc 𝐴 ∈ 𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 ∩ cint 4946 Oncon0 6384 suc csuc 6386 Inaccwcwina 10722 Inacccina 10723 |
| 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: gruex 44317 |
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