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Theorem rankeq1o 36153
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 8525 . . . . . . 7 1o ≠ ∅
2 neeq1 3001 . . . . . . 7 ((rank‘𝐴) = 1o → ((rank‘𝐴) ≠ ∅ ↔ 1o ≠ ∅))
31, 2mpbiri 258 . . . . . 6 ((rank‘𝐴) = 1o → (rank‘𝐴) ≠ ∅)
43neneqd 2943 . . . . 5 ((rank‘𝐴) = 1o → ¬ (rank‘𝐴) = ∅)
5 fvprc 6899 . . . . 5 𝐴 ∈ V → (rank‘𝐴) = ∅)
64, 5nsyl2 141 . . . 4 ((rank‘𝐴) = 1o𝐴 ∈ V)
7 fveqeq2 6916 . . . . . 6 (𝑥 = 𝐴 → ((rank‘𝑥) = 1o ↔ (rank‘𝐴) = 1o))
8 eqeq1 2739 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 1o𝐴 = 1o))
97, 8imbi12d 344 . . . . 5 (𝑥 = 𝐴 → (((rank‘𝑥) = 1o𝑥 = 1o) ↔ ((rank‘𝐴) = 1o𝐴 = 1o)))
10 neeq1 3001 . . . . . . . 8 ((rank‘𝑥) = 1o → ((rank‘𝑥) ≠ ∅ ↔ 1o ≠ ∅))
111, 10mpbiri 258 . . . . . . 7 ((rank‘𝑥) = 1o → (rank‘𝑥) ≠ ∅)
12 vex 3482 . . . . . . . . 9 𝑥 ∈ V
1312rankeq0 9899 . . . . . . . 8 (𝑥 = ∅ ↔ (rank‘𝑥) = ∅)
1413necon3bii 2991 . . . . . . 7 (𝑥 ≠ ∅ ↔ (rank‘𝑥) ≠ ∅)
1511, 14sylibr 234 . . . . . 6 ((rank‘𝑥) = 1o𝑥 ≠ ∅)
1612rankval 9854 . . . . . . . 8 (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}
1716eqeq1i 2740 . . . . . . 7 ((rank‘𝑥) = 1o {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
18 ssrab2 4090 . . . . . . . . . . 11 {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
19 elirr 9635 . . . . . . . . . . . . . 14 ¬ 1o ∈ 1o
20 1oex 8515 . . . . . . . . . . . . . . 15 1o ∈ V
21 id 22 . . . . . . . . . . . . . . 15 (V = 1o → V = 1o)
2220, 21eleqtrid 2845 . . . . . . . . . . . . . 14 (V = 1o → 1o ∈ 1o)
2319, 22mto 197 . . . . . . . . . . . . 13 ¬ V = 1o
24 inteq 4954 . . . . . . . . . . . . . . 15 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅)
25 int0 4967 . . . . . . . . . . . . . . 15 ∅ = V
2624, 25eqtrdi 2791 . . . . . . . . . . . . . 14 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = V)
2726eqeq1d 2737 . . . . . . . . . . . . 13 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o ↔ V = 1o))
2823, 27mtbiri 327 . . . . . . . . . . . 12 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = ∅ → ¬ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o)
2928necon2ai 2968 . . . . . . . . . . 11 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
30 onint 7810 . . . . . . . . . . 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 587 . . . . . . . . . 10 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
32 eleq1 2827 . . . . . . . . . 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 232 . . . . . . . . 9 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → 1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
34 suceq 6452 . . . . . . . . . . . . 13 (𝑦 = 1o → suc 𝑦 = suc 1o)
3534fveq2d 6911 . . . . . . . . . . . 12 (𝑦 = 1o → (𝑅1‘suc 𝑦) = (𝑅1‘suc 1o))
36 df-1o 8505 . . . . . . . . . . . . . . . . 17 1o = suc ∅
3736fveq2i 6910 . . . . . . . . . . . . . . . 16 (𝑅1‘1o) = (𝑅1‘suc ∅)
38 0elon 6440 . . . . . . . . . . . . . . . . 17 ∅ ∈ On
39 r1suc 9808 . . . . . . . . . . . . . . . . 17 (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅))
4038, 39ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)
41 r10 9806 . . . . . . . . . . . . . . . . 17 (𝑅1‘∅) = ∅
4241pweqi 4621 . . . . . . . . . . . . . . . 16 𝒫 (𝑅1‘∅) = 𝒫 ∅
4337, 40, 423eqtri 2767 . . . . . . . . . . . . . . 15 (𝑅1‘1o) = 𝒫 ∅
4443pweqi 4621 . . . . . . . . . . . . . 14 𝒫 (𝑅1‘1o) = 𝒫 𝒫 ∅
45 pw0 4817 . . . . . . . . . . . . . . 15 𝒫 ∅ = {∅}
4645pweqi 4621 . . . . . . . . . . . . . 14 𝒫 𝒫 ∅ = 𝒫 {∅}
47 pwpw0 4818 . . . . . . . . . . . . . 14 𝒫 {∅} = {∅, {∅}}
4844, 46, 473eqtrri 2768 . . . . . . . . . . . . 13 {∅, {∅}} = 𝒫 (𝑅1‘1o)
49 1on 8517 . . . . . . . . . . . . . 14 1o ∈ On
50 r1suc 9808 . . . . . . . . . . . . . 14 (1o ∈ On → (𝑅1‘suc 1o) = 𝒫 (𝑅1‘1o))
5149, 50ax-mp 5 . . . . . . . . . . . . 13 (𝑅1‘suc 1o) = 𝒫 (𝑅1‘1o)
5248, 51eqtr4i 2766 . . . . . . . . . . . 12 {∅, {∅}} = (𝑅1‘suc 1o)
5335, 52eqtr4di 2793 . . . . . . . . . . 11 (𝑦 = 1o → (𝑅1‘suc 𝑦) = {∅, {∅}})
5453eleq2d 2825 . . . . . . . . . 10 (𝑦 = 1o → (𝑥 ∈ (𝑅1‘suc 𝑦) ↔ 𝑥 ∈ {∅, {∅}}))
5554elrab 3695 . . . . . . . . 9 (1o ∈ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
5633, 55sylib 218 . . . . . . . 8 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}))
5712elpr 4655 . . . . . . . . . 10 (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
58 df-ne 2939 . . . . . . . . . . . 12 (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
59 orel1 888 . . . . . . . . . . . 12 𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
6058, 59sylbi 217 . . . . . . . . . . 11 (𝑥 ≠ ∅ → ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 = {∅}))
61 df1o2 8512 . . . . . . . . . . . . 13 1o = {∅}
62 eqeq2 2747 . . . . . . . . . . . . 13 (𝑥 = {∅} → (1o = 𝑥 ↔ 1o = {∅}))
6361, 62mpbiri 258 . . . . . . . . . . . 12 (𝑥 = {∅} → 1o = 𝑥)
6463eqcomd 2741 . . . . . . . . . . 11 (𝑥 = {∅} → 𝑥 = 1o)
6560, 64syl6com 37 . . . . . . . . . 10 ((𝑥 = ∅ ∨ 𝑥 = {∅}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
6657, 65sylbi 217 . . . . . . . . 9 (𝑥 ∈ {∅, {∅}} → (𝑥 ≠ ∅ → 𝑥 = 1o))
6766adantl 481 . . . . . . . 8 ((1o ∈ On ∧ 𝑥 ∈ {∅, {∅}}) → (𝑥 ≠ ∅ → 𝑥 = 1o))
6856, 67syl 17 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
6917, 68sylbi 217 . . . . . 6 ((rank‘𝑥) = 1o → (𝑥 ≠ ∅ → 𝑥 = 1o))
7015, 69mpd 15 . . . . 5 ((rank‘𝑥) = 1o𝑥 = 1o)
719, 70vtoclg 3554 . . . 4 (𝐴 ∈ V → ((rank‘𝐴) = 1o𝐴 = 1o))
726, 71mpcom 38 . . 3 ((rank‘𝐴) = 1o𝐴 = 1o)
73 fveq2 6907 . . . 4 (𝐴 = 1o → (rank‘𝐴) = (rank‘1o))
74 r111 9813 . . . . . . 7 𝑅1:On–1-1→V
75 f1dm 6809 . . . . . . 7 (𝑅1:On–1-1→V → dom 𝑅1 = On)
7674, 75ax-mp 5 . . . . . 6 dom 𝑅1 = On
7749, 76eleqtrri 2838 . . . . 5 1o ∈ dom 𝑅1
78 rankonid 9867 . . . . 5 (1o ∈ dom 𝑅1 ↔ (rank‘1o) = 1o)
7977, 78mpbi 230 . . . 4 (rank‘1o) = 1o
8073, 79eqtrdi 2791 . . 3 (𝐴 = 1o → (rank‘𝐴) = 1o)
8172, 80impbii 209 . 2 ((rank‘𝐴) = 1o𝐴 = 1o)
8261eqeq2i 2748 . 2 (𝐴 = 1o𝐴 = {∅})
8381, 82bitri 275 1 ((rank‘𝐴) = 1o𝐴 = {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  {crab 3433  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631  {cpr 4633   cint 4951  dom cdm 5689  Oncon0 6386  suc csuc 6388  1-1wf1 6560  cfv 6563  1oc1o 8498  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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679
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-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-r1 9802  df-rank 9803
This theorem is referenced by: (None)
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