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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 9817 | . . . . . . . 8 ⊢ Tr (𝑅1‘suc (rank‘𝐴)) | |
| 2 | rankidb 9841 | . . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) | |
| 3 | trss 5269 | . . . . . . . 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 9842 | . . . . . . . 8 ⊢ (rank‘𝐴) ∈ dom 𝑅1 | |
| 6 | r1sucg 9810 | . . . . . . . 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 4023 | . . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴))) | 
| 9 | sspwuni 5099 | . . . . . 6 ⊢ (𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ ∪ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
| 10 | 8, 9 | sylib 218 | . . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | 
| 11 | fvex 6918 | . . . . . 6 ⊢ (𝑅1‘(rank‘𝐴)) ∈ V | |
| 12 | 11 | elpw2 5333 | . . . . 5 ⊢ (∪ 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ ∪ 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | 
| 13 | 10, 12 | sylibr 234 | . . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴))) | 
| 14 | 13, 7 | eleqtrrdi 2851 | . . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) | 
| 15 | r1elwf 9837 | . . 3 ⊢ (∪ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On)) | |
| 16 | 14, 15 | syl 17 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On)) | 
| 17 | pwwf 9848 | . . 3 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 ∪ 𝐴 ∈ ∪ (𝑅1 “ On)) | |
| 18 | pwuni 4944 | . . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
| 19 | sswf 9849 | . . . 4 ⊢ ((𝒫 ∪ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On)) | |
| 20 | 18, 19 | mpan2 691 | . . 3 ⊢ (𝒫 ∪ 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On)) | 
| 21 | 17, 20 | sylbi 217 | . 2 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On)) | 
| 22 | 16, 21 | impbii 209 | 1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝐴 ∈ ∪ (𝑅1 “ On)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 𝒫 cpw 4599 ∪ cuni 4906 Tr wtr 5258 dom cdm 5684 “ cima 5687 Oncon0 6383 suc csuc 6385 ‘cfv 6560 𝑅1cr1 9803 rankcrnk 9804 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-r1 9805 df-rank 9806 | 
| This theorem is referenced by: rankuni2b 9894 r1limwun 10777 wfgru 10857 dmwf 44987 rnwf 44988 wfaxun 45021 wfac8prim 45024 | 
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