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Mirrors > Home > MPE Home > Th. List > uniwf | Structured version Visualization version GIF version |
Description: A union is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
uniwf | ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝐴 ∈ ∪ (𝑅1 “ On)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1tr 9191 | . . . . . . . 8 ⊢ Tr (𝑅1‘suc (rank‘𝐴)) | |
2 | rankidb 9215 | . . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) | |
3 | trss 5167 | . . . . . . . 8 ⊢ (Tr (𝑅1‘suc (rank‘𝐴)) → (𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))) | |
4 | 1, 2, 3 | mpsyl 68 | . . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴))) |
5 | rankdmr1 9216 | . . . . . . . 8 ⊢ (rank‘𝐴) ∈ dom 𝑅1 | |
6 | r1sucg 9184 | . . . . . . . 8 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))) | |
7 | 5, 6 | ax-mp 5 | . . . . . . 7 ⊢ (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)) |
8 | 4, 7 | sseqtrdi 4005 | . . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴))) |
9 | sspwuni 5008 | . . . . . 6 ⊢ (𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ ∪ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
10 | 8, 9 | sylib 220 | . . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) |
11 | fvex 6669 | . . . . . 6 ⊢ (𝑅1‘(rank‘𝐴)) ∈ V | |
12 | 11 | elpw2 5234 | . . . . 5 ⊢ (∪ 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ ∪ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) |
13 | 10, 12 | sylibr 236 | . . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴))) |
14 | 13, 7 | eleqtrrdi 2924 | . . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) |
15 | r1elwf 9211 | . . 3 ⊢ (∪ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On)) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On)) |
17 | pwwf 9222 | . . 3 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 ∪ 𝐴 ∈ ∪ (𝑅1 “ On)) | |
18 | pwuni 4861 | . . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
19 | sswf 9223 | . . . 4 ⊢ ((𝒫 ∪ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On)) | |
20 | 18, 19 | mpan2 689 | . . 3 ⊢ (𝒫 ∪ 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
21 | 17, 20 | sylbi 219 | . 2 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
22 | 16, 21 | impbii 211 | 1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝐴 ∈ ∪ (𝑅1 “ On)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ⊆ wss 3924 𝒫 cpw 4525 ∪ cuni 4824 Tr wtr 5158 dom cdm 5541 “ cima 5544 Oncon0 6177 suc csuc 6179 ‘cfv 6341 𝑅1cr1 9177 rankcrnk 9178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-om 7567 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-r1 9179 df-rank 9180 |
This theorem is referenced by: rankuni2b 9268 r1limwun 10144 wfgru 10224 |
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