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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 9710 | . . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
2 | 1 | simpri 487 | . . . . 5 ⊢ Lim dom 𝑅1 |
3 | limord 6381 | . . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
4 | ordsson 7721 | . . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On) | |
5 | 2, 3, 4 | mp2b 10 | . . . 4 ⊢ dom 𝑅1 ⊆ On |
6 | elfvdm 6883 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1) | |
7 | 5, 6 | sselid 3946 | . . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ On) |
8 | r1tr 9720 | . . . . . 6 ⊢ Tr (𝑅1‘𝐵) | |
9 | trss 5237 | . . . . . 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 4567 | . . . . 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 9713 | . . . . 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 2837 | . . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ (𝑅1‘suc 𝐵)) |
16 | suceq 6387 | . . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) | |
17 | 16 | fveq2d 6850 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑅1‘suc 𝑥) = (𝑅1‘suc 𝐵)) |
18 | 17 | eleq2d 2820 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc 𝐵))) |
19 | 18 | rspcev 3583 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) |
20 | 7, 15, 19 | syl2anc 585 | . 2 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) |
21 | rankwflemb 9737 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) | |
22 | 20, 21 | sylibr 233 | 1 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∃wrex 3070 ⊆ wss 3914 𝒫 cpw 4564 ∪ cuni 4869 Tr wtr 5226 dom cdm 5637 “ cima 5640 Ord word 6320 Oncon0 6321 Lim wlim 6322 suc csuc 6323 Fun wfun 6494 ‘cfv 6500 𝑅1cr1 9706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-r1 9708 |
This theorem is referenced by: rankr1ai 9742 pwwf 9751 sswf 9752 unwf 9754 uniwf 9763 rankonidlem 9772 r1pw 9789 r1pwcl 9791 rankr1id 9806 tcrank 9828 dfac12lem2 10088 r1limwun 10680 r1wunlim 10681 inatsk 10722 |
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