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Theorem rankeq1o 33153
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 7923 . . . . . . 7 1o ≠ ∅
2 neeq1 3029 . . . . . . 7 ((rank‘𝐴) = 1o → ((rank‘𝐴) ≠ ∅ ↔ 1o ≠ ∅))
31, 2mpbiri 250 . . . . . 6 ((rank‘𝐴) = 1o → (rank‘𝐴) ≠ ∅)
43neneqd 2972 . . . . 5 ((rank‘𝐴) = 1o → ¬ (rank‘𝐴) = ∅)
5 fvprc 6494 . . . . 5 𝐴 ∈ V → (rank‘𝐴) = ∅)
64, 5nsyl2 145 . . . 4 ((rank‘𝐴) = 1o𝐴 ∈ V)
7 fveqeq2 6510 . . . . . 6 (𝑥 = 𝐴 → ((rank‘𝑥) = 1o ↔ (rank‘𝐴) = 1o))
8 eqeq1 2782 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 1o𝐴 = 1o))
97, 8imbi12d 337 . . . . 5 (𝑥 = 𝐴 → (((rank‘𝑥) = 1o𝑥 = 1o) ↔ ((rank‘𝐴) = 1o𝐴 = 1o)))
10 neeq1 3029 . . . . . . . 8 ((rank‘𝑥) = 1o → ((rank‘𝑥) ≠ ∅ ↔ 1o ≠ ∅))
111, 10mpbiri 250 . . . . . . 7 ((rank‘𝑥) = 1o → (rank‘𝑥) ≠ ∅)
12 vex 3418 . . . . . . . . 9 𝑥 ∈ V
1312rankeq0 9086 . . . . . . . 8 (𝑥 = ∅ ↔ (rank‘𝑥) = ∅)
1413necon3bii 3019 . . . . . . 7 (𝑥 ≠ ∅ ↔ (rank‘𝑥) ≠ ∅)
1511, 14sylibr 226 . . . . . 6 ((rank‘𝑥) = 1o𝑥 ≠ ∅)
1612rankval 9041 . . . . . . . 8 (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}
1716eqeq1i 2783 . . . . . . 7 ((rank‘𝑥) = 1o {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
18 ssrab2 3948 . . . . . . . . . . 11 {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
19 elirr 8858 . . . . . . . . . . . . . 14 ¬ 1o ∈ 1o
20 1oex 7915 . . . . . . . . . . . . . . 15 1o ∈ V
21 id 22 . . . . . . . . . . . . . . 15 (V = 1o → V = 1o)
2220, 21syl5eleq 2872 . . . . . . . . . . . . . 14 (V = 1o → 1o ∈ 1o)
2319, 22mto 189 . . . . . . . . . . . . 13 ¬ V = 1o
24 inteq 4753 . . . . . . . . . . . . . . 15 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅)
25 int0 4764 . . . . . . . . . . . . . . 15 ∅ = V
2624, 25syl6eq 2830 . . . . . . . . . . . . . 14 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = V)
2726eqeq1d 2780 . . . . . . . . . . . . 13 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o ↔ V = 1o))
2823, 27mtbiri 319 . . . . . . . . . . . 12 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ¬ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
2928necon2ai 2996 . . . . . . . . . . 11 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
30 onint 7328 . . . . . . . . . . 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 578 . . . . . . . . . 10 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
32 eleq1 2853 . . . . . . . . . 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 224 . . . . . . . . 9 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → 1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
34 suceq 6096 . . . . . . . . . . . . 13 (𝑦 = 1o → suc 𝑦 = suc 1o)
3534fveq2d 6505 . . . . . . . . . . . 12 (𝑦 = 1o → (𝑅1‘suc 𝑦) = (𝑅1‘suc 1o))
36 df-1o 7907 . . . . . . . . . . . . . . . . 17 1o = suc ∅
3736fveq2i 6504 . . . . . . . . . . . . . . . 16 (𝑅1‘1o) = (𝑅1‘suc ∅)
38 0elon 6084 . . . . . . . . . . . . . . . . 17 ∅ ∈ On
39 r1suc 8995 . . . . . . . . . . . . . . . . 17 (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅))
4038, 39ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)
41 r10 8993 . . . . . . . . . . . . . . . . 17 (𝑅1‘∅) = ∅
4241pweqi 4427 . . . . . . . . . . . . . . . 16 𝒫 (𝑅1‘∅) = 𝒫 ∅
4337, 40, 423eqtri 2806 . . . . . . . . . . . . . . 15 (𝑅1‘1o) = 𝒫 ∅
4443pweqi 4427 . . . . . . . . . . . . . 14 𝒫 (𝑅1‘1o) = 𝒫 𝒫 ∅
45 pw0 4620 . . . . . . . . . . . . . . 15 𝒫 ∅ = {∅}
4645pweqi 4427 . . . . . . . . . . . . . 14 𝒫 𝒫 ∅ = 𝒫 {∅}
47 pwpw0 4621 . . . . . . . . . . . . . 14 𝒫 {∅} = {∅, {∅}}
4844, 46, 473eqtrri 2807 . . . . . . . . . . . . 13 {∅, {∅}} = 𝒫 (𝑅1‘1o)
49 1on 7914 . . . . . . . . . . . . . 14 1o ∈ On
50 r1suc 8995 . . . . . . . . . . . . . 14 (1o ∈ On → (𝑅1‘suc 1o) = 𝒫 (𝑅1‘1o))
5149, 50ax-mp 5 . . . . . . . . . . . . 13 (𝑅1‘suc 1o) = 𝒫 (𝑅1‘1o)
5248, 51eqtr4i 2805 . . . . . . . . . . . 12 {∅, {∅}} = (𝑅1‘suc 1o)
5335, 52syl6eqr 2832 . . . . . . . . . . 11 (𝑦 = 1o → (𝑅1‘suc 𝑦) = {∅, {∅}})
5453eleq2d 2851 . . . . . . . . . 10 (𝑦 = 1o → (𝑥 ∈ (𝑅1‘suc 𝑦) ↔ 𝑥 ∈ {∅, {∅}}))
5554elrab 3595 . . . . . . . . 9 (1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
5633, 55sylib 210 . . . . . . . 8 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
5712elpr 4465 . . . . . . . . . 10 (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
58 df-ne 2968 . . . . . . . . . . . 12 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
59 orel1 872 . . . . . . . . . . . 12 𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
6058, 59sylbi 209 . . . . . . . . . . 11 (𝑥 ≠ ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
61 df1o2 7920 . . . . . . . . . . . . 13 1o = {∅}
62 eqeq2 2789 . . . . . . . . . . . . 13 (𝑥 = {∅} → (1o = 𝑥 ↔ 1o = {∅}))
6361, 62mpbiri 250 . . . . . . . . . . . 12 (𝑥 = {∅} → 1o = 𝑥)
6463eqcomd 2784 . . . . . . . . . . 11 (𝑥 = {∅} → 𝑥 = 1o)
6560, 64syl6com 37 . . . . . . . . . 10 ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
6657, 65sylbi 209 . . . . . . . . 9 (𝑥 ∈ {∅, {∅}} → (𝑥 ≠ ∅ → 𝑥 = 1o))
6766adantl 474 . . . . . . . 8 ((1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
6856, 67syl 17 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
6917, 68sylbi 209 . . . . . 6 ((rank‘𝑥) = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
7015, 69mpd 15 . . . . 5 ((rank‘𝑥) = 1o𝑥 = 1o)
719, 70vtoclg 3486 . . . 4 (𝐴 ∈ V → ((rank‘𝐴) = 1o𝐴 = 1o))
726, 71mpcom 38 . . 3 ((rank‘𝐴) = 1o𝐴 = 1o)
73 fveq2 6501 . . . 4 (𝐴 = 1o → (rank‘𝐴) = (rank‘1o))
74 r111 9000 . . . . . . 7 𝑅1:On–1-1→V
75 f1dm 6410 . . . . . . 7 (𝑅1:On–1-1→V → dom 𝑅1 = On)
7674, 75ax-mp 5 . . . . . 6 dom 𝑅1 = On
7749, 76eleqtrri 2865 . . . . 5 1o ∈ dom 𝑅1
78 rankonid 9054 . . . . 5 (1o ∈ dom 𝑅1 ↔ (rank‘1o) = 1o)
7977, 78mpbi 222 . . . 4 (rank‘1o) = 1o
8073, 79syl6eq 2830 . . 3 (𝐴 = 1o → (rank‘𝐴) = 1o)
8172, 80impbii 201 . 2 ((rank‘𝐴) = 1o𝐴 = 1o)
8261eqeq2i 2790 . 2 (𝐴 = 1o𝐴 = {∅})
8381, 82bitri 267 1 ((rank‘𝐴) = 1o𝐴 = {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wo 833   = wceq 1507  wcel 2050  wne 2967  {crab 3092  Vcvv 3415  wss 3831  c0 4180  𝒫 cpw 4423  {csn 4442  {cpr 4444   cint 4750  dom cdm 5408  Oncon0 6031  suc csuc 6033  1-1wf1 6187  cfv 6190  1oc1o 7900  𝑅1cr1 8987  rankcrnk 8988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-reg 8853  ax-inf2 8900
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-om 7399  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-1o 7907  df-er 8091  df-en 8309  df-dom 8310  df-sdom 8311  df-r1 8989  df-rank 8990
This theorem is referenced by: (None)
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