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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 8906 | . . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
2 | 1 | simpri 481 | . . . . 5 ⊢ Lim dom 𝑅1 |
3 | limord 6022 | . . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
4 | ordsson 7250 | . . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On) | |
5 | 2, 3, 4 | mp2b 10 | . . . 4 ⊢ dom 𝑅1 ⊆ On |
6 | elfvdm 6465 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1) | |
7 | 5, 6 | sseldi 3825 | . . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ On) |
8 | r1tr 8916 | . . . . . 6 ⊢ Tr (𝑅1‘𝐵) | |
9 | trss 4984 | . . . . . 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 4386 | . . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝐴 ∈ 𝒫 (𝑅1‘𝐵) ↔ 𝐴 ⊆ (𝑅1‘𝐵))) | |
12 | 10, 11 | mpbird 249 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ 𝒫 (𝑅1‘𝐵)) |
13 | r1sucg 8909 | . . . . 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 2909 | . . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ (𝑅1‘suc 𝐵)) |
16 | suceq 6028 | . . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) | |
17 | 16 | fveq2d 6437 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑅1‘suc 𝑥) = (𝑅1‘suc 𝐵)) |
18 | 17 | eleq2d 2892 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc 𝐵))) |
19 | 18 | rspcev 3526 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) |
20 | 7, 15, 19 | syl2anc 579 | . 2 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) |
21 | rankwflemb 8933 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) | |
22 | 20, 21 | sylibr 226 | 1 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 ⊆ wss 3798 𝒫 cpw 4378 ∪ cuni 4658 Tr wtr 4975 dom cdm 5342 “ cima 5345 Ord word 5962 Oncon0 5963 Lim wlim 5964 suc csuc 5965 Fun wfun 6117 ‘cfv 6123 𝑅1cr1 8902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-r1 8904 |
This theorem is referenced by: rankr1ai 8938 pwwf 8947 sswf 8948 unwf 8950 uniwf 8959 rankonidlem 8968 r1pw 8985 r1pwcl 8987 rankr1id 9002 tcrank 9024 dfac12lem2 9281 r1limwun 9873 r1wunlim 9874 inatsk 9915 |
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