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| Mirrors > Home > MPE Home > Th. List > r1elwf | Structured version Visualization version GIF version | ||
| Description: Any member of the cumulative hierarchy is well-founded. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
| Ref | Expression |
|---|---|
| r1elwf | ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r1funlim 9806 | . . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
| 2 | 1 | simpri 485 | . . . . 5 ⊢ Lim dom 𝑅1 |
| 3 | limord 6444 | . . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
| 4 | ordsson 7803 | . . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On) | |
| 5 | 2, 3, 4 | mp2b 10 | . . . 4 ⊢ dom 𝑅1 ⊆ On |
| 6 | elfvdm 6943 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1) | |
| 7 | 5, 6 | sselid 3981 | . . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ On) |
| 8 | r1tr 9816 | . . . . . 6 ⊢ Tr (𝑅1‘𝐵) | |
| 9 | trss 5270 | . . . . . 6 ⊢ (Tr (𝑅1‘𝐵) → (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ⊆ (𝑅1‘𝐵))) | |
| 10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ⊆ (𝑅1‘𝐵)) |
| 11 | elpwg 4603 | . . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝐴 ∈ 𝒫 (𝑅1‘𝐵) ↔ 𝐴 ⊆ (𝑅1‘𝐵))) | |
| 12 | 10, 11 | mpbird 257 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ 𝒫 (𝑅1‘𝐵)) |
| 13 | r1sucg 9809 | . . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → (𝑅1‘suc 𝐵) = 𝒫 (𝑅1‘𝐵)) | |
| 14 | 6, 13 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝑅1‘suc 𝐵) = 𝒫 (𝑅1‘𝐵)) |
| 15 | 12, 14 | eleqtrrd 2844 | . . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ (𝑅1‘suc 𝐵)) |
| 16 | suceq 6450 | . . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) | |
| 17 | 16 | fveq2d 6910 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑅1‘suc 𝑥) = (𝑅1‘suc 𝐵)) |
| 18 | 17 | eleq2d 2827 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc 𝐵))) |
| 19 | 18 | rspcev 3622 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) |
| 20 | 7, 15, 19 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) |
| 21 | rankwflemb 9833 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) | |
| 22 | 20, 21 | sylibr 234 | 1 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 Tr wtr 5259 dom cdm 5685 “ cima 5688 Ord word 6383 Oncon0 6384 Lim wlim 6385 suc csuc 6386 Fun wfun 6555 ‘cfv 6561 𝑅1cr1 9802 |
| 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-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-ral 3062 df-rex 3071 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-iun 4993 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-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-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-r1 9804 |
| This theorem is referenced by: rankr1ai 9838 pwwf 9847 sswf 9848 unwf 9850 uniwf 9859 rankonidlem 9868 r1pw 9885 r1pwcl 9887 rankr1id 9902 tcrank 9924 dfac12lem2 10185 r1limwun 10776 r1wunlim 10777 inatsk 10818 |
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