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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 6645 | . . 3 ⊢ (𝐴 = ∅ → (rank‘𝐴) = (rank‘∅)) | |
2 | r1funlim 9179 | . . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
3 | 2 | simpri 489 | . . . . . 6 ⊢ Lim dom 𝑅1 |
4 | limomss 7565 | . . . . . 6 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ ω ⊆ dom 𝑅1 |
6 | peano1 7581 | . . . . 5 ⊢ ∅ ∈ ω | |
7 | 5, 6 | sselii 3912 | . . . 4 ⊢ ∅ ∈ dom 𝑅1 |
8 | rankonid 9242 | . . . 4 ⊢ (∅ ∈ dom 𝑅1 ↔ (rank‘∅) = ∅) | |
9 | 7, 8 | mpbi 233 | . . 3 ⊢ (rank‘∅) = ∅ |
10 | 1, 9 | eqtrdi 2849 | . 2 ⊢ (𝐴 = ∅ → (rank‘𝐴) = ∅) |
11 | eqimss 3971 | . . . . . . 7 ⊢ ((rank‘𝐴) = ∅ → (rank‘𝐴) ⊆ ∅) | |
12 | 11 | adantl 485 | . . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → (rank‘𝐴) ⊆ ∅) |
13 | simpl 486 | . . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ∈ ∪ (𝑅1 “ On)) | |
14 | rankr1bg 9216 | . . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∅ ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘∅) ↔ (rank‘𝐴) ⊆ ∅)) | |
15 | 13, 7, 14 | sylancl 589 | . . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → (𝐴 ⊆ (𝑅1‘∅) ↔ (rank‘𝐴) ⊆ ∅)) |
16 | 12, 15 | mpbird 260 | . . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ⊆ (𝑅1‘∅)) |
17 | r10 9181 | . . . . 5 ⊢ (𝑅1‘∅) = ∅ | |
18 | 16, 17 | sseqtrdi 3965 | . . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ⊆ ∅) |
19 | ss0 4306 | . . . 4 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 = ∅) |
21 | 20 | ex 416 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ((rank‘𝐴) = ∅ → 𝐴 = ∅)) |
22 | 10, 21 | impbid2 229 | 1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 dom cdm 5519 “ cima 5522 Oncon0 6159 Lim wlim 6160 Fun wfun 6318 ‘cfv 6324 ωcom 7560 𝑅1cr1 9175 rankcrnk 9176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-r1 9177 df-rank 9178 |
This theorem is referenced by: rankeq0 9274 |
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