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Theorem rankeq1o 35800
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 8515 . . . . . . 7 1o ≠ ∅
2 neeq1 3000 . . . . . . 7 ((rank‘𝐴) = 1o → ((rank‘𝐴) ≠ ∅ ↔ 1o ≠ ∅))
31, 2mpbiri 257 . . . . . 6 ((rank‘𝐴) = 1o → (rank‘𝐴) ≠ ∅)
43neneqd 2942 . . . . 5 ((rank‘𝐴) = 1o → ¬ (rank‘𝐴) = ∅)
5 fvprc 6894 . . . . 5 𝐴 ∈ V → (rank‘𝐴) = ∅)
64, 5nsyl2 141 . . . 4 ((rank‘𝐴) = 1o𝐴 ∈ V)
7 fveqeq2 6911 . . . . . 6 (𝑥 = 𝐴 → ((rank‘𝑥) = 1o ↔ (rank‘𝐴) = 1o))
8 eqeq1 2732 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 1o𝐴 = 1o))
97, 8imbi12d 343 . . . . 5 (𝑥 = 𝐴 → (((rank‘𝑥) = 1o𝑥 = 1o) ↔ ((rank‘𝐴) = 1o𝐴 = 1o)))
10 neeq1 3000 . . . . . . . 8 ((rank‘𝑥) = 1o → ((rank‘𝑥) ≠ ∅ ↔ 1o ≠ ∅))
111, 10mpbiri 257 . . . . . . 7 ((rank‘𝑥) = 1o → (rank‘𝑥) ≠ ∅)
12 vex 3477 . . . . . . . . 9 𝑥 ∈ V
1312rankeq0 9892 . . . . . . . 8 (𝑥 = ∅ ↔ (rank‘𝑥) = ∅)
1413necon3bii 2990 . . . . . . 7 (𝑥 ≠ ∅ ↔ (rank‘𝑥) ≠ ∅)
1511, 14sylibr 233 . . . . . 6 ((rank‘𝑥) = 1o𝑥 ≠ ∅)
1612rankval 9847 . . . . . . . 8 (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}
1716eqeq1i 2733 . . . . . . 7 ((rank‘𝑥) = 1o {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
18 ssrab2 4077 . . . . . . . . . . 11 {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
19 elirr 9628 . . . . . . . . . . . . . 14 ¬ 1o ∈ 1o
20 1oex 8503 . . . . . . . . . . . . . . 15 1o ∈ V
21 id 22 . . . . . . . . . . . . . . 15 (V = 1o → V = 1o)
2220, 21eleqtrid 2835 . . . . . . . . . . . . . 14 (V = 1o → 1o ∈ 1o)
2319, 22mto 196 . . . . . . . . . . . . 13 ¬ V = 1o
24 inteq 4956 . . . . . . . . . . . . . . 15 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅)
25 int0 4969 . . . . . . . . . . . . . . 15 ∅ = V
2624, 25eqtrdi 2784 . . . . . . . . . . . . . 14 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = V)
2726eqeq1d 2730 . . . . . . . . . . . . 13 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o ↔ V = 1o))
2823, 27mtbiri 326 . . . . . . . . . . . 12 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ¬ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
2928necon2ai 2967 . . . . . . . . . . 11 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
30 onint 7799 . . . . . . . . . . 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 585 . . . . . . . . . 10 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
32 eleq1 2817 . . . . . . . . . 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 231 . . . . . . . . 9 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → 1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
34 suceq 6440 . . . . . . . . . . . . 13 (𝑦 = 1o → suc 𝑦 = suc 1o)
3534fveq2d 6906 . . . . . . . . . . . 12 (𝑦 = 1o → (𝑅1‘suc 𝑦) = (𝑅1‘suc 1o))
36 df-1o 8493 . . . . . . . . . . . . . . . . 17 1o = suc ∅
3736fveq2i 6905 . . . . . . . . . . . . . . . 16 (𝑅1‘1o) = (𝑅1‘suc ∅)
38 0elon 6428 . . . . . . . . . . . . . . . . 17 ∅ ∈ On
39 r1suc 9801 . . . . . . . . . . . . . . . . 17 (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅))
4038, 39ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)
41 r10 9799 . . . . . . . . . . . . . . . . 17 (𝑅1‘∅) = ∅
4241pweqi 4622 . . . . . . . . . . . . . . . 16 𝒫 (𝑅1‘∅) = 𝒫 ∅
4337, 40, 423eqtri 2760 . . . . . . . . . . . . . . 15 (𝑅1‘1o) = 𝒫 ∅
4443pweqi 4622 . . . . . . . . . . . . . 14 𝒫 (𝑅1‘1o) = 𝒫 𝒫 ∅
45 pw0 4820 . . . . . . . . . . . . . . 15 𝒫 ∅ = {∅}
4645pweqi 4622 . . . . . . . . . . . . . 14 𝒫 𝒫 ∅ = 𝒫 {∅}
47 pwpw0 4821 . . . . . . . . . . . . . 14 𝒫 {∅} = {∅, {∅}}
4844, 46, 473eqtrri 2761 . . . . . . . . . . . . 13 {∅, {∅}} = 𝒫 (𝑅1‘1o)
49 1on 8505 . . . . . . . . . . . . . 14 1o ∈ On
50 r1suc 9801 . . . . . . . . . . . . . 14 (1o ∈ On → (𝑅1‘suc 1o) = 𝒫 (𝑅1‘1o))
5149, 50ax-mp 5 . . . . . . . . . . . . 13 (𝑅1‘suc 1o) = 𝒫 (𝑅1‘1o)
5248, 51eqtr4i 2759 . . . . . . . . . . . 12 {∅, {∅}} = (𝑅1‘suc 1o)
5335, 52eqtr4di 2786 . . . . . . . . . . 11 (𝑦 = 1o → (𝑅1‘suc 𝑦) = {∅, {∅}})
5453eleq2d 2815 . . . . . . . . . 10 (𝑦 = 1o → (𝑥 ∈ (𝑅1‘suc 𝑦) ↔ 𝑥 ∈ {∅, {∅}}))
5554elrab 3684 . . . . . . . . 9 (1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
5633, 55sylib 217 . . . . . . . 8 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
5712elpr 4656 . . . . . . . . . 10 (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
58 df-ne 2938 . . . . . . . . . . . 12 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
59 orel1 886 . . . . . . . . . . . 12 𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
6058, 59sylbi 216 . . . . . . . . . . 11 (𝑥 ≠ ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
61 df1o2 8500 . . . . . . . . . . . . 13 1o = {∅}
62 eqeq2 2740 . . . . . . . . . . . . 13 (𝑥 = {∅} → (1o = 𝑥 ↔ 1o = {∅}))
6361, 62mpbiri 257 . . . . . . . . . . . 12 (𝑥 = {∅} → 1o = 𝑥)
6463eqcomd 2734 . . . . . . . . . . 11 (𝑥 = {∅} → 𝑥 = 1o)
6560, 64syl6com 37 . . . . . . . . . 10 ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
6657, 65sylbi 216 . . . . . . . . 9 (𝑥 ∈ {∅, {∅}} → (𝑥 ≠ ∅ → 𝑥 = 1o))
6766adantl 480 . . . . . . . 8 ((1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
6856, 67syl 17 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
6917, 68sylbi 216 . . . . . 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 6902 . . . 4 (𝐴 = 1o → (rank‘𝐴) = (rank‘1o))
74 r111 9806 . . . . . . 7 𝑅1:On–1-1→V
75 f1dm 6802 . . . . . . 7 (𝑅1:On–1-1→V → dom 𝑅1 = On)
7674, 75ax-mp 5 . . . . . 6 dom 𝑅1 = On
7749, 76eleqtrri 2828 . . . . 5 1o ∈ dom 𝑅1
78 rankonid 9860 . . . . 5 (1o ∈ dom 𝑅1 ↔ (rank‘1o) = 1o)
7977, 78mpbi 229 . . . 4 (rank‘1o) = 1o
8073, 79eqtrdi 2784 . . 3 (𝐴 = 1o → (rank‘𝐴) = 1o)
8172, 80impbii 208 . 2 ((rank‘𝐴) = 1o𝐴 = 1o)
8261eqeq2i 2741 . 2 (𝐴 = 1o𝐴 = {∅})
8381, 82bitri 274 1 ((rank‘𝐴) = 1o𝐴 = {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  wne 2937  {crab 3430  Vcvv 3473  wss 3949  c0 4326  𝒫 cpw 4606  {csn 4632  {cpr 4634   cint 4953  dom cdm 5682  Oncon0 6374  suc csuc 6376  1-1wf1 6550  cfv 6553  1oc1o 8486  𝑅1cr1 9793  rankcrnk 9794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9623  ax-inf2 9672
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-r1 9795  df-rank 9796
This theorem is referenced by: (None)
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