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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 6907 | . . 3 ⊢ (𝐴 = ∅ → (rank‘𝐴) = (rank‘∅)) | |
2 | r1funlim 9804 | . . . . . . 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 7911 | . . . . 5 ⊢ ∅ ∈ ω | |
7 | 5, 6 | sselii 3992 | . . . 4 ⊢ ∅ ∈ dom 𝑅1 |
8 | rankonid 9867 | . . . 4 ⊢ (∅ ∈ dom 𝑅1 ↔ (rank‘∅) = ∅) | |
9 | 7, 8 | mpbi 230 | . . 3 ⊢ (rank‘∅) = ∅ |
10 | 1, 9 | eqtrdi 2791 | . 2 ⊢ (𝐴 = ∅ → (rank‘𝐴) = ∅) |
11 | eqimss 4054 | . . . . . . 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 9841 | . . . . . . 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 9806 | . . . . 5 ⊢ (𝑅1‘∅) = ∅ | |
18 | 16, 17 | sseqtrdi 4046 | . . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = ∅) → 𝐴 ⊆ ∅) |
19 | ss0 4408 | . . . 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 1537 ∈ wcel 2106 ⊆ wss 3963 ∅c0 4339 ∪ cuni 4912 dom cdm 5689 “ cima 5692 Oncon0 6386 Lim wlim 6387 Fun wfun 6557 ‘cfv 6563 ωcom 7887 𝑅1cr1 9800 rankcrnk 9801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-r1 9802 df-rank 9803 |
This theorem is referenced by: rankeq0 9899 |
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