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Theorem rankeq1o 32722
Description: The only set with rank 1𝑜 is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)
Assertion
Ref Expression
rankeq1o ((rank‘𝐴) = 1𝑜𝐴 = {∅})

Proof of Theorem rankeq1o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 7780 . . . . . . 7 1𝑜 ≠ ∅
2 neeq1 2999 . . . . . . 7 ((rank‘𝐴) = 1𝑜 → ((rank‘𝐴) ≠ ∅ ↔ 1𝑜 ≠ ∅))
31, 2mpbiri 249 . . . . . 6 ((rank‘𝐴) = 1𝑜 → (rank‘𝐴) ≠ ∅)
43neneqd 2942 . . . . 5 ((rank‘𝐴) = 1𝑜 → ¬ (rank‘𝐴) = ∅)
5 fvprc 6368 . . . . 5 𝐴 ∈ V → (rank‘𝐴) = ∅)
64, 5nsyl2 144 . . . 4 ((rank‘𝐴) = 1𝑜𝐴 ∈ V)
7 fveqeq2 6384 . . . . . 6 (𝑥 = 𝐴 → ((rank‘𝑥) = 1𝑜 ↔ (rank‘𝐴) = 1𝑜))
8 eqeq1 2769 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 1𝑜𝐴 = 1𝑜))
97, 8imbi12d 335 . . . . 5 (𝑥 = 𝐴 → (((rank‘𝑥) = 1𝑜𝑥 = 1𝑜) ↔ ((rank‘𝐴) = 1𝑜𝐴 = 1𝑜)))
10 neeq1 2999 . . . . . . . 8 ((rank‘𝑥) = 1𝑜 → ((rank‘𝑥) ≠ ∅ ↔ 1𝑜 ≠ ∅))
111, 10mpbiri 249 . . . . . . 7 ((rank‘𝑥) = 1𝑜 → (rank‘𝑥) ≠ ∅)
12 vex 3353 . . . . . . . . 9 𝑥 ∈ V
1312rankeq0 8939 . . . . . . . 8 (𝑥 = ∅ ↔ (rank‘𝑥) = ∅)
1413necon3bii 2989 . . . . . . 7 (𝑥 ≠ ∅ ↔ (rank‘𝑥) ≠ ∅)
1511, 14sylibr 225 . . . . . 6 ((rank‘𝑥) = 1𝑜𝑥 ≠ ∅)
1612rankval 8894 . . . . . . . 8 (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}
1716eqeq1i 2770 . . . . . . 7 ((rank‘𝑥) = 1𝑜 {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜)
18 ssrab2 3847 . . . . . . . . . . 11 {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
19 elirr 8709 . . . . . . . . . . . . . 14 ¬ 1𝑜 ∈ 1𝑜
20 1oex 7772 . . . . . . . . . . . . . . 15 1𝑜 ∈ V
21 id 22 . . . . . . . . . . . . . . 15 (V = 1𝑜 → V = 1𝑜)
2220, 21syl5eleq 2850 . . . . . . . . . . . . . 14 (V = 1𝑜 → 1𝑜 ∈ 1𝑜)
2319, 22mto 188 . . . . . . . . . . . . 13 ¬ V = 1𝑜
24 inteq 4636 . . . . . . . . . . . . . . 15 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅)
25 int0 4647 . . . . . . . . . . . . . . 15 ∅ = V
2624, 25syl6eq 2815 . . . . . . . . . . . . . 14 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = V)
2726eqeq1d 2767 . . . . . . . . . . . . 13 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 ↔ V = 1𝑜))
2823, 27mtbiri 318 . . . . . . . . . . . 12 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ¬ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜)
2928necon2ai 2966 . . . . . . . . . . 11 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
30 onint 7193 . . . . . . . . . . 11 (({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅) → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
3118, 29, 30sylancr 581 . . . . . . . . . 10 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
32 eleq1 2832 . . . . . . . . . 10 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ 1𝑜 ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}))
3331, 32mpbid 223 . . . . . . . . 9 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → 1𝑜 ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
34 suceq 5973 . . . . . . . . . . . . 13 (𝑦 = 1𝑜 → suc 𝑦 = suc 1𝑜)
3534fveq2d 6379 . . . . . . . . . . . 12 (𝑦 = 1𝑜 → (𝑅1‘suc 𝑦) = (𝑅1‘suc 1𝑜))
36 df-1o 7764 . . . . . . . . . . . . . . . . 17 1𝑜 = suc ∅
3736fveq2i 6378 . . . . . . . . . . . . . . . 16 (𝑅1‘1𝑜) = (𝑅1‘suc ∅)
38 0elon 5961 . . . . . . . . . . . . . . . . 17 ∅ ∈ On
39 r1suc 8848 . . . . . . . . . . . . . . . . 17 (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅))
4038, 39ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)
41 r10 8846 . . . . . . . . . . . . . . . . 17 (𝑅1‘∅) = ∅
4241pweqi 4319 . . . . . . . . . . . . . . . 16 𝒫 (𝑅1‘∅) = 𝒫 ∅
4337, 40, 423eqtri 2791 . . . . . . . . . . . . . . 15 (𝑅1‘1𝑜) = 𝒫 ∅
4443pweqi 4319 . . . . . . . . . . . . . 14 𝒫 (𝑅1‘1𝑜) = 𝒫 𝒫 ∅
45 pw0 4497 . . . . . . . . . . . . . . 15 𝒫 ∅ = {∅}
4645pweqi 4319 . . . . . . . . . . . . . 14 𝒫 𝒫 ∅ = 𝒫 {∅}
47 pwpw0 4498 . . . . . . . . . . . . . 14 𝒫 {∅} = {∅, {∅}}
4844, 46, 473eqtrri 2792 . . . . . . . . . . . . 13 {∅, {∅}} = 𝒫 (𝑅1‘1𝑜)
49 1on 7771 . . . . . . . . . . . . . 14 1𝑜 ∈ On
50 r1suc 8848 . . . . . . . . . . . . . 14 (1𝑜 ∈ On → (𝑅1‘suc 1𝑜) = 𝒫 (𝑅1‘1𝑜))
5149, 50ax-mp 5 . . . . . . . . . . . . 13 (𝑅1‘suc 1𝑜) = 𝒫 (𝑅1‘1𝑜)
5248, 51eqtr4i 2790 . . . . . . . . . . . 12 {∅, {∅}} = (𝑅1‘suc 1𝑜)
5335, 52syl6eqr 2817 . . . . . . . . . . 11 (𝑦 = 1𝑜 → (𝑅1‘suc 𝑦) = {∅, {∅}})
5453eleq2d 2830 . . . . . . . . . 10 (𝑦 = 1𝑜 → (𝑥 ∈ (𝑅1‘suc 𝑦) ↔ 𝑥 ∈ {∅, {∅}}))
5554elrab 3519 . . . . . . . . 9 (1𝑜 ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ (1𝑜 ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
5633, 55sylib 209 . . . . . . . 8 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → (1𝑜 ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
5712elpr 4357 . . . . . . . . . 10 (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
58 df-ne 2938 . . . . . . . . . . . 12 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
59 orel1 912 . . . . . . . . . . . 12 𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
6058, 59sylbi 208 . . . . . . . . . . 11 (𝑥 ≠ ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
61 df1o2 7777 . . . . . . . . . . . . 13 1𝑜 = {∅}
62 eqeq2 2776 . . . . . . . . . . . . 13 (𝑥 = {∅} → (1𝑜 = 𝑥 ↔ 1𝑜 = {∅}))
6361, 62mpbiri 249 . . . . . . . . . . . 12 (𝑥 = {∅} → 1𝑜 = 𝑥)
6463eqcomd 2771 . . . . . . . . . . 11 (𝑥 = {∅} → 𝑥 = 1𝑜)
6560, 64syl6com 37 . . . . . . . . . 10 ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 ≠ ∅ → 𝑥 = 1𝑜))
6657, 65sylbi 208 . . . . . . . . 9 (𝑥 ∈ {∅, {∅}} → (𝑥 ≠ ∅ → 𝑥 = 1𝑜))
6766adantl 473 . . . . . . . 8 ((1𝑜 ∈ On ∧ 𝑥 ∈ {∅, {∅}}) → (𝑥 ≠ ∅ → 𝑥 = 1𝑜))
6856, 67syl 17 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1𝑜 → (𝑥 ≠ ∅ → 𝑥 = 1𝑜))
6917, 68sylbi 208 . . . . . 6 ((rank‘𝑥) = 1𝑜 → (𝑥 ≠ ∅ → 𝑥 = 1𝑜))
7015, 69mpd 15 . . . . 5 ((rank‘𝑥) = 1𝑜𝑥 = 1𝑜)
719, 70vtoclg 3418 . . . 4 (𝐴 ∈ V → ((rank‘𝐴) = 1𝑜𝐴 = 1𝑜))
726, 71mpcom 38 . . 3 ((rank‘𝐴) = 1𝑜𝐴 = 1𝑜)
73 fveq2 6375 . . . 4 (𝐴 = 1𝑜 → (rank‘𝐴) = (rank‘1𝑜))
74 r111 8853 . . . . . . 7 𝑅1:On–1-1→V
75 f1dm 6287 . . . . . . 7 (𝑅1:On–1-1→V → dom 𝑅1 = On)
7674, 75ax-mp 5 . . . . . 6 dom 𝑅1 = On
7749, 76eleqtrri 2843 . . . . 5 1𝑜 ∈ dom 𝑅1
78 rankonid 8907 . . . . 5 (1𝑜 ∈ dom 𝑅1 ↔ (rank‘1𝑜) = 1𝑜)
7977, 78mpbi 221 . . . 4 (rank‘1𝑜) = 1𝑜
8073, 79syl6eq 2815 . . 3 (𝐴 = 1𝑜 → (rank‘𝐴) = 1𝑜)
8172, 80impbii 200 . 2 ((rank‘𝐴) = 1𝑜𝐴 = 1𝑜)
8261eqeq2i 2777 . 2 (𝐴 = 1𝑜𝐴 = {∅})
8381, 82bitri 266 1 ((rank‘𝐴) = 1𝑜𝐴 = {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  wne 2937  {crab 3059  Vcvv 3350  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334  {cpr 4336   cint 4633  dom cdm 5277  Oncon0 5908  suc csuc 5910  1-1wf1 6065  cfv 6068  1𝑜c1o 7757  𝑅1cr1 8840  rankcrnk 8841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-reg 8704  ax-inf2 8753
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-r1 8842  df-rank 8843
This theorem is referenced by: (None)
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