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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 6664 | . . 3 ⊢ (𝐴 = ∅ → (rank‘𝐴) = (rank‘∅)) | |
2 | r1funlim 9184 | . . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
3 | 2 | simpri 486 | . . . . . 6 ⊢ Lim dom 𝑅1 |
4 | limomss 7573 | . . . . . 6 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ ω ⊆ dom 𝑅1 |
6 | peano1 7589 | . . . . 5 ⊢ ∅ ∈ ω | |
7 | 5, 6 | sselii 3963 | . . . 4 ⊢ ∅ ∈ dom 𝑅1 |
8 | rankonid 9247 | . . . 4 ⊢ (∅ ∈ dom 𝑅1 ↔ (rank‘∅) = ∅) | |
9 | 7, 8 | mpbi 231 | . . 3 ⊢ (rank‘∅) = ∅ |
10 | 1, 9 | syl6eq 2872 | . 2 ⊢ (𝐴 = ∅ → (rank‘𝐴) = ∅) |
11 | eqimss 4022 | . . . . . . 7 ⊢ ((rank‘𝐴) = ∅ → (rank‘𝐴) ⊆ ∅) | |
12 | 11 | adantl 482 | . . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → (rank‘𝐴) ⊆ ∅) |
13 | simpl 483 | . . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ∈ ∪ (𝑅1 “ On)) | |
14 | rankr1bg 9221 | . . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∅ ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘∅) ↔ (rank‘𝐴) ⊆ ∅)) | |
15 | 13, 7, 14 | sylancl 586 | . . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → (𝐴 ⊆ (𝑅1‘∅) ↔ (rank‘𝐴) ⊆ ∅)) |
16 | 12, 15 | mpbird 258 | . . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ⊆ (𝑅1‘∅)) |
17 | r10 9186 | . . . . 5 ⊢ (𝑅1‘∅) = ∅ | |
18 | 16, 17 | sseqtrdi 4016 | . . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ⊆ ∅) |
19 | ss0 4351 | . . . 4 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 = ∅) |
21 | 20 | ex 413 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ((rank‘𝐴) = ∅ → 𝐴 = ∅)) |
22 | 10, 21 | impbid2 227 | 1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4832 dom cdm 5549 “ cima 5552 Oncon0 6185 Lim wlim 6186 Fun wfun 6343 ‘cfv 6349 ωcom 7568 𝑅1cr1 9180 rankcrnk 9181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-r1 9182 df-rank 9183 |
This theorem is referenced by: rankeq0 9279 |
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