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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 6906 | . . 3 ⊢ (𝐴 = ∅ → (rank‘𝐴) = (rank‘∅)) | |
| 2 | r1funlim 9806 | . . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
| 3 | 2 | simpri 485 | . . . . . 6 ⊢ Lim dom 𝑅1 |
| 4 | limomss 7892 | . . . . . 6 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
| 5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ ω ⊆ dom 𝑅1 |
| 6 | peano1 7910 | . . . . 5 ⊢ ∅ ∈ ω | |
| 7 | 5, 6 | sselii 3980 | . . . 4 ⊢ ∅ ∈ dom 𝑅1 |
| 8 | rankonid 9869 | . . . 4 ⊢ (∅ ∈ dom 𝑅1 ↔ (rank‘∅) = ∅) | |
| 9 | 7, 8 | mpbi 230 | . . 3 ⊢ (rank‘∅) = ∅ |
| 10 | 1, 9 | eqtrdi 2793 | . 2 ⊢ (𝐴 = ∅ → (rank‘𝐴) = ∅) |
| 11 | eqimss 4042 | . . . . . . 7 ⊢ ((rank‘𝐴) = ∅ → (rank‘𝐴) ⊆ ∅) | |
| 12 | 11 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → (rank‘𝐴) ⊆ ∅) |
| 13 | simpl 482 | . . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ∈ ∪ (𝑅1 “ On)) | |
| 14 | rankr1bg 9843 | . . . . . . 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 257 | . . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ⊆ (𝑅1‘∅)) |
| 17 | r10 9808 | . . . . 5 ⊢ (𝑅1‘∅) = ∅ | |
| 18 | 16, 17 | sseqtrdi 4024 | . . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ⊆ ∅) |
| 19 | ss0 4402 | . . . 4 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
| 20 | 18, 19 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 = ∅) |
| 21 | 20 | ex 412 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ((rank‘𝐴) = ∅ → 𝐴 = ∅)) |
| 22 | 10, 21 | impbid2 226 | 1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 dom cdm 5685 “ cima 5688 Oncon0 6384 Lim wlim 6385 Fun wfun 6555 ‘cfv 6561 ωcom 7887 𝑅1cr1 9802 rankcrnk 9803 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-r1 9804 df-rank 9805 |
| This theorem is referenced by: rankeq0 9901 |
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