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Theorem rankeq1o 33706
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 8106 . . . . . . 7 1o ≠ ∅
2 neeq1 3073 . . . . . . 7 ((rank‘𝐴) = 1o → ((rank‘𝐴) ≠ ∅ ↔ 1o ≠ ∅))
31, 2mpbiri 261 . . . . . 6 ((rank‘𝐴) = 1o → (rank‘𝐴) ≠ ∅)
43neneqd 3016 . . . . 5 ((rank‘𝐴) = 1o → ¬ (rank‘𝐴) = ∅)
5 fvprc 6645 . . . . 5 𝐴 ∈ V → (rank‘𝐴) = ∅)
64, 5nsyl2 143 . . . 4 ((rank‘𝐴) = 1o𝐴 ∈ V)
7 fveqeq2 6661 . . . . . 6 (𝑥 = 𝐴 → ((rank‘𝑥) = 1o ↔ (rank‘𝐴) = 1o))
8 eqeq1 2826 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 1o𝐴 = 1o))
97, 8imbi12d 348 . . . . 5 (𝑥 = 𝐴 → (((rank‘𝑥) = 1o𝑥 = 1o) ↔ ((rank‘𝐴) = 1o𝐴 = 1o)))
10 neeq1 3073 . . . . . . . 8 ((rank‘𝑥) = 1o → ((rank‘𝑥) ≠ ∅ ↔ 1o ≠ ∅))
111, 10mpbiri 261 . . . . . . 7 ((rank‘𝑥) = 1o → (rank‘𝑥) ≠ ∅)
12 vex 3472 . . . . . . . . 9 𝑥 ∈ V
1312rankeq0 9278 . . . . . . . 8 (𝑥 = ∅ ↔ (rank‘𝑥) = ∅)
1413necon3bii 3063 . . . . . . 7 (𝑥 ≠ ∅ ↔ (rank‘𝑥) ≠ ∅)
1511, 14sylibr 237 . . . . . 6 ((rank‘𝑥) = 1o𝑥 ≠ ∅)
1612rankval 9233 . . . . . . . 8 (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}
1716eqeq1i 2827 . . . . . . 7 ((rank‘𝑥) = 1o {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
18 ssrab2 4031 . . . . . . . . . . 11 {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
19 elirr 9049 . . . . . . . . . . . . . 14 ¬ 1o ∈ 1o
20 1oex 8097 . . . . . . . . . . . . . . 15 1o ∈ V
21 id 22 . . . . . . . . . . . . . . 15 (V = 1o → V = 1o)
2220, 21eleqtrid 2920 . . . . . . . . . . . . . 14 (V = 1o → 1o ∈ 1o)
2319, 22mto 200 . . . . . . . . . . . . 13 ¬ V = 1o
24 inteq 4854 . . . . . . . . . . . . . . 15 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅)
25 int0 4865 . . . . . . . . . . . . . . 15 ∅ = V
2624, 25syl6eq 2873 . . . . . . . . . . . . . 14 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = V)
2726eqeq1d 2824 . . . . . . . . . . . . 13 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o ↔ V = 1o))
2823, 27mtbiri 330 . . . . . . . . . . . 12 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ¬ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
2928necon2ai 3040 . . . . . . . . . . 11 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
30 onint 7495 . . . . . . . . . . 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 590 . . . . . . . . . 10 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
32 eleq1 2901 . . . . . . . . . 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 235 . . . . . . . . 9 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → 1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
34 suceq 6234 . . . . . . . . . . . . 13 (𝑦 = 1o → suc 𝑦 = suc 1o)
3534fveq2d 6656 . . . . . . . . . . . 12 (𝑦 = 1o → (𝑅1‘suc 𝑦) = (𝑅1‘suc 1o))
36 df-1o 8089 . . . . . . . . . . . . . . . . 17 1o = suc ∅
3736fveq2i 6655 . . . . . . . . . . . . . . . 16 (𝑅1‘1o) = (𝑅1‘suc ∅)
38 0elon 6222 . . . . . . . . . . . . . . . . 17 ∅ ∈ On
39 r1suc 9187 . . . . . . . . . . . . . . . . 17 (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅))
4038, 39ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)
41 r10 9185 . . . . . . . . . . . . . . . . 17 (𝑅1‘∅) = ∅
4241pweqi 4529 . . . . . . . . . . . . . . . 16 𝒫 (𝑅1‘∅) = 𝒫 ∅
4337, 40, 423eqtri 2849 . . . . . . . . . . . . . . 15 (𝑅1‘1o) = 𝒫 ∅
4443pweqi 4529 . . . . . . . . . . . . . 14 𝒫 (𝑅1‘1o) = 𝒫 𝒫 ∅
45 pw0 4718 . . . . . . . . . . . . . . 15 𝒫 ∅ = {∅}
4645pweqi 4529 . . . . . . . . . . . . . 14 𝒫 𝒫 ∅ = 𝒫 {∅}
47 pwpw0 4719 . . . . . . . . . . . . . 14 𝒫 {∅} = {∅, {∅}}
4844, 46, 473eqtrri 2850 . . . . . . . . . . . . 13 {∅, {∅}} = 𝒫 (𝑅1‘1o)
49 1on 8096 . . . . . . . . . . . . . 14 1o ∈ On
50 r1suc 9187 . . . . . . . . . . . . . 14 (1o ∈ On → (𝑅1‘suc 1o) = 𝒫 (𝑅1‘1o))
5149, 50ax-mp 5 . . . . . . . . . . . . 13 (𝑅1‘suc 1o) = 𝒫 (𝑅1‘1o)
5248, 51eqtr4i 2848 . . . . . . . . . . . 12 {∅, {∅}} = (𝑅1‘suc 1o)
5335, 52eqtr4di 2875 . . . . . . . . . . 11 (𝑦 = 1o → (𝑅1‘suc 𝑦) = {∅, {∅}})
5453eleq2d 2899 . . . . . . . . . 10 (𝑦 = 1o → (𝑥 ∈ (𝑅1‘suc 𝑦) ↔ 𝑥 ∈ {∅, {∅}}))
5554elrab 3655 . . . . . . . . 9 (1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
5633, 55sylib 221 . . . . . . . 8 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
5712elpr 4562 . . . . . . . . . 10 (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
58 df-ne 3012 . . . . . . . . . . . 12 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
59 orel1 886 . . . . . . . . . . . 12 𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
6058, 59sylbi 220 . . . . . . . . . . 11 (𝑥 ≠ ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
61 df1o2 8103 . . . . . . . . . . . . 13 1o = {∅}
62 eqeq2 2834 . . . . . . . . . . . . 13 (𝑥 = {∅} → (1o = 𝑥 ↔ 1o = {∅}))
6361, 62mpbiri 261 . . . . . . . . . . . 12 (𝑥 = {∅} → 1o = 𝑥)
6463eqcomd 2828 . . . . . . . . . . 11 (𝑥 = {∅} → 𝑥 = 1o)
6560, 64syl6com 37 . . . . . . . . . 10 ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
6657, 65sylbi 220 . . . . . . . . 9 (𝑥 ∈ {∅, {∅}} → (𝑥 ≠ ∅ → 𝑥 = 1o))
6766adantl 485 . . . . . . . 8 ((1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
6856, 67syl 17 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
6917, 68sylbi 220 . . . . . 6 ((rank‘𝑥) = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
7015, 69mpd 15 . . . . 5 ((rank‘𝑥) = 1o𝑥 = 1o)
719, 70vtoclg 3542 . . . 4 (𝐴 ∈ V → ((rank‘𝐴) = 1o𝐴 = 1o))
726, 71mpcom 38 . . 3 ((rank‘𝐴) = 1o𝐴 = 1o)
73 fveq2 6652 . . . 4 (𝐴 = 1o → (rank‘𝐴) = (rank‘1o))
74 r111 9192 . . . . . . 7 𝑅1:On–1-1→V
75 f1dm 6560 . . . . . . 7 (𝑅1:On–1-1→V → dom 𝑅1 = On)
7674, 75ax-mp 5 . . . . . 6 dom 𝑅1 = On
7749, 76eleqtrri 2913 . . . . 5 1o ∈ dom 𝑅1
78 rankonid 9246 . . . . 5 (1o ∈ dom 𝑅1 ↔ (rank‘1o) = 1o)
7977, 78mpbi 233 . . . 4 (rank‘1o) = 1o
8073, 79syl6eq 2873 . . 3 (𝐴 = 1o → (rank‘𝐴) = 1o)
8172, 80impbii 212 . 2 ((rank‘𝐴) = 1o𝐴 = 1o)
8261eqeq2i 2835 . 2 (𝐴 = 1o𝐴 = {∅})
8381, 82bitri 278 1 ((rank‘𝐴) = 1o𝐴 = {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2114  wne 3011  {crab 3134  Vcvv 3469  wss 3908  c0 4265  𝒫 cpw 4511  {csn 4539  {cpr 4541   cint 4851  dom cdm 5532  Oncon0 6169  suc csuc 6171  1-1wf1 6331  cfv 6334  1oc1o 8082  𝑅1cr1 9179  rankcrnk 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-r1 9181  df-rank 9182
This theorem is referenced by: (None)
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