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Mirrors > Home > MPE Home > Th. List > rankeq0b | Structured version Visualization version GIF version |
Description: A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankeq0b | ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6191 | . . 3 ⊢ (𝐴 = ∅ → (rank‘𝐴) = (rank‘∅)) | |
2 | r1funlim 8629 | . . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
3 | 2 | simpri 478 | . . . . . 6 ⊢ Lim dom 𝑅1 |
4 | limomss 7070 | . . . . . 6 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ ω ⊆ dom 𝑅1 |
6 | peano1 7085 | . . . . 5 ⊢ ∅ ∈ ω | |
7 | 5, 6 | sselii 3600 | . . . 4 ⊢ ∅ ∈ dom 𝑅1 |
8 | rankonid 8692 | . . . 4 ⊢ (∅ ∈ dom 𝑅1 ↔ (rank‘∅) = ∅) | |
9 | 7, 8 | mpbi 220 | . . 3 ⊢ (rank‘∅) = ∅ |
10 | 1, 9 | syl6eq 2672 | . 2 ⊢ (𝐴 = ∅ → (rank‘𝐴) = ∅) |
11 | eqimss 3657 | . . . . . . 7 ⊢ ((rank‘𝐴) = ∅ → (rank‘𝐴) ⊆ ∅) | |
12 | 11 | adantl 482 | . . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → (rank‘𝐴) ⊆ ∅) |
13 | simpl 473 | . . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ∈ ∪ (𝑅1 “ On)) | |
14 | rankr1bg 8666 | . . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∅ ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘∅) ↔ (rank‘𝐴) ⊆ ∅)) | |
15 | 13, 7, 14 | sylancl 694 | . . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → (𝐴 ⊆ (𝑅1‘∅) ↔ (rank‘𝐴) ⊆ ∅)) |
16 | 12, 15 | mpbird 247 | . . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ⊆ (𝑅1‘∅)) |
17 | r10 8631 | . . . . 5 ⊢ (𝑅1‘∅) = ∅ | |
18 | 16, 17 | syl6sseq 3651 | . . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ⊆ ∅) |
19 | ss0 3974 | . . . 4 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 = ∅) |
21 | 20 | ex 450 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ((rank‘𝐴) = ∅ → 𝐴 = ∅)) |
22 | 10, 21 | impbid2 216 | 1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ∅c0 3915 ∪ cuni 4436 dom cdm 5114 “ cima 5117 Oncon0 5723 Lim wlim 5724 Fun wfun 5882 ‘cfv 5888 ωcom 7065 𝑅1cr1 8625 rankcrnk 8626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-r1 8627 df-rank 8628 |
This theorem is referenced by: rankeq0 8724 |
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