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Theorem rankeq1o 35676
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 ((rankβ€˜π΄) = 1o ↔ 𝐴 = {βˆ…})

Proof of Theorem rankeq1o
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 8489 . . . . . . 7 1o β‰  βˆ…
2 neeq1 2997 . . . . . . 7 ((rankβ€˜π΄) = 1o β†’ ((rankβ€˜π΄) β‰  βˆ… ↔ 1o β‰  βˆ…))
31, 2mpbiri 258 . . . . . 6 ((rankβ€˜π΄) = 1o β†’ (rankβ€˜π΄) β‰  βˆ…)
43neneqd 2939 . . . . 5 ((rankβ€˜π΄) = 1o β†’ Β¬ (rankβ€˜π΄) = βˆ…)
5 fvprc 6877 . . . . 5 (Β¬ 𝐴 ∈ V β†’ (rankβ€˜π΄) = βˆ…)
64, 5nsyl2 141 . . . 4 ((rankβ€˜π΄) = 1o β†’ 𝐴 ∈ V)
7 fveqeq2 6894 . . . . . 6 (π‘₯ = 𝐴 β†’ ((rankβ€˜π‘₯) = 1o ↔ (rankβ€˜π΄) = 1o))
8 eqeq1 2730 . . . . . 6 (π‘₯ = 𝐴 β†’ (π‘₯ = 1o ↔ 𝐴 = 1o))
97, 8imbi12d 344 . . . . 5 (π‘₯ = 𝐴 β†’ (((rankβ€˜π‘₯) = 1o β†’ π‘₯ = 1o) ↔ ((rankβ€˜π΄) = 1o β†’ 𝐴 = 1o)))
10 neeq1 2997 . . . . . . . 8 ((rankβ€˜π‘₯) = 1o β†’ ((rankβ€˜π‘₯) β‰  βˆ… ↔ 1o β‰  βˆ…))
111, 10mpbiri 258 . . . . . . 7 ((rankβ€˜π‘₯) = 1o β†’ (rankβ€˜π‘₯) β‰  βˆ…)
12 vex 3472 . . . . . . . . 9 π‘₯ ∈ V
1312rankeq0 9858 . . . . . . . 8 (π‘₯ = βˆ… ↔ (rankβ€˜π‘₯) = βˆ…)
1413necon3bii 2987 . . . . . . 7 (π‘₯ β‰  βˆ… ↔ (rankβ€˜π‘₯) β‰  βˆ…)
1511, 14sylibr 233 . . . . . 6 ((rankβ€˜π‘₯) = 1o β†’ π‘₯ β‰  βˆ…)
1612rankval 9813 . . . . . . . 8 (rankβ€˜π‘₯) = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)}
1716eqeq1i 2731 . . . . . . 7 ((rankβ€˜π‘₯) = 1o ↔ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o)
18 ssrab2 4072 . . . . . . . . . . 11 {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} βŠ† On
19 elirr 9594 . . . . . . . . . . . . . 14 Β¬ 1o ∈ 1o
20 1oex 8477 . . . . . . . . . . . . . . 15 1o ∈ V
21 id 22 . . . . . . . . . . . . . . 15 (V = 1o β†’ V = 1o)
2220, 21eleqtrid 2833 . . . . . . . . . . . . . 14 (V = 1o β†’ 1o ∈ 1o)
2319, 22mto 196 . . . . . . . . . . . . 13 Β¬ V = 1o
24 inteq 4946 . . . . . . . . . . . . . . 15 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = βˆ… β†’ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = ∩ βˆ…)
25 int0 4959 . . . . . . . . . . . . . . 15 ∩ βˆ… = V
2624, 25eqtrdi 2782 . . . . . . . . . . . . . 14 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = βˆ… β†’ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = V)
2726eqeq1d 2728 . . . . . . . . . . . . 13 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = βˆ… β†’ (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o ↔ V = 1o))
2823, 27mtbiri 327 . . . . . . . . . . . 12 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = βˆ… β†’ Β¬ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o)
2928necon2ai 2964 . . . . . . . . . . 11 (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o β†’ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} β‰  βˆ…)
30 onint 7775 . . . . . . . . . . 11 (({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} βŠ† On ∧ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} β‰  βˆ…) β†’ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ∈ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)})
3118, 29, 30sylancr 586 . . . . . . . . . 10 (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o β†’ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ∈ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)})
32 eleq1 2815 . . . . . . . . . 10 (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o β†’ (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ∈ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ↔ 1o ∈ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)}))
3331, 32mpbid 231 . . . . . . . . 9 (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o β†’ 1o ∈ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)})
34 suceq 6424 . . . . . . . . . . . . 13 (𝑦 = 1o β†’ suc 𝑦 = suc 1o)
3534fveq2d 6889 . . . . . . . . . . . 12 (𝑦 = 1o β†’ (𝑅1β€˜suc 𝑦) = (𝑅1β€˜suc 1o))
36 df-1o 8467 . . . . . . . . . . . . . . . . 17 1o = suc βˆ…
3736fveq2i 6888 . . . . . . . . . . . . . . . 16 (𝑅1β€˜1o) = (𝑅1β€˜suc βˆ…)
38 0elon 6412 . . . . . . . . . . . . . . . . 17 βˆ… ∈ On
39 r1suc 9767 . . . . . . . . . . . . . . . . 17 (βˆ… ∈ On β†’ (𝑅1β€˜suc βˆ…) = 𝒫 (𝑅1β€˜βˆ…))
4038, 39ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑅1β€˜suc βˆ…) = 𝒫 (𝑅1β€˜βˆ…)
41 r10 9765 . . . . . . . . . . . . . . . . 17 (𝑅1β€˜βˆ…) = βˆ…
4241pweqi 4613 . . . . . . . . . . . . . . . 16 𝒫 (𝑅1β€˜βˆ…) = 𝒫 βˆ…
4337, 40, 423eqtri 2758 . . . . . . . . . . . . . . 15 (𝑅1β€˜1o) = 𝒫 βˆ…
4443pweqi 4613 . . . . . . . . . . . . . 14 𝒫 (𝑅1β€˜1o) = 𝒫 𝒫 βˆ…
45 pw0 4810 . . . . . . . . . . . . . . 15 𝒫 βˆ… = {βˆ…}
4645pweqi 4613 . . . . . . . . . . . . . 14 𝒫 𝒫 βˆ… = 𝒫 {βˆ…}
47 pwpw0 4811 . . . . . . . . . . . . . 14 𝒫 {βˆ…} = {βˆ…, {βˆ…}}
4844, 46, 473eqtrri 2759 . . . . . . . . . . . . 13 {βˆ…, {βˆ…}} = 𝒫 (𝑅1β€˜1o)
49 1on 8479 . . . . . . . . . . . . . 14 1o ∈ On
50 r1suc 9767 . . . . . . . . . . . . . 14 (1o ∈ On β†’ (𝑅1β€˜suc 1o) = 𝒫 (𝑅1β€˜1o))
5149, 50ax-mp 5 . . . . . . . . . . . . 13 (𝑅1β€˜suc 1o) = 𝒫 (𝑅1β€˜1o)
5248, 51eqtr4i 2757 . . . . . . . . . . . 12 {βˆ…, {βˆ…}} = (𝑅1β€˜suc 1o)
5335, 52eqtr4di 2784 . . . . . . . . . . 11 (𝑦 = 1o β†’ (𝑅1β€˜suc 𝑦) = {βˆ…, {βˆ…}})
5453eleq2d 2813 . . . . . . . . . 10 (𝑦 = 1o β†’ (π‘₯ ∈ (𝑅1β€˜suc 𝑦) ↔ π‘₯ ∈ {βˆ…, {βˆ…}}))
5554elrab 3678 . . . . . . . . 9 (1o ∈ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ↔ (1o ∈ On ∧ π‘₯ ∈ {βˆ…, {βˆ…}}))
5633, 55sylib 217 . . . . . . . 8 (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o β†’ (1o ∈ On ∧ π‘₯ ∈ {βˆ…, {βˆ…}}))
5712elpr 4646 . . . . . . . . . 10 (π‘₯ ∈ {βˆ…, {βˆ…}} ↔ (π‘₯ = βˆ… ∨ π‘₯ = {βˆ…}))
58 df-ne 2935 . . . . . . . . . . . 12 (π‘₯ β‰  βˆ… ↔ Β¬ π‘₯ = βˆ…)
59 orel1 885 . . . . . . . . . . . 12 (Β¬ π‘₯ = βˆ… β†’ ((π‘₯ = βˆ… ∨ π‘₯ = {βˆ…}) β†’ π‘₯ = {βˆ…}))
6058, 59sylbi 216 . . . . . . . . . . 11 (π‘₯ β‰  βˆ… β†’ ((π‘₯ = βˆ… ∨ π‘₯ = {βˆ…}) β†’ π‘₯ = {βˆ…}))
61 df1o2 8474 . . . . . . . . . . . . 13 1o = {βˆ…}
62 eqeq2 2738 . . . . . . . . . . . . 13 (π‘₯ = {βˆ…} β†’ (1o = π‘₯ ↔ 1o = {βˆ…}))
6361, 62mpbiri 258 . . . . . . . . . . . 12 (π‘₯ = {βˆ…} β†’ 1o = π‘₯)
6463eqcomd 2732 . . . . . . . . . . 11 (π‘₯ = {βˆ…} β†’ π‘₯ = 1o)
6560, 64syl6com 37 . . . . . . . . . 10 ((π‘₯ = βˆ… ∨ π‘₯ = {βˆ…}) β†’ (π‘₯ β‰  βˆ… β†’ π‘₯ = 1o))
6657, 65sylbi 216 . . . . . . . . 9 (π‘₯ ∈ {βˆ…, {βˆ…}} β†’ (π‘₯ β‰  βˆ… β†’ π‘₯ = 1o))
6766adantl 481 . . . . . . . 8 ((1o ∈ On ∧ π‘₯ ∈ {βˆ…, {βˆ…}}) β†’ (π‘₯ β‰  βˆ… β†’ π‘₯ = 1o))
6856, 67syl 17 . . . . . . 7 (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o β†’ (π‘₯ β‰  βˆ… β†’ π‘₯ = 1o))
6917, 68sylbi 216 . . . . . 6 ((rankβ€˜π‘₯) = 1o β†’ (π‘₯ β‰  βˆ… β†’ π‘₯ = 1o))
7015, 69mpd 15 . . . . 5 ((rankβ€˜π‘₯) = 1o β†’ π‘₯ = 1o)
719, 70vtoclg 3537 . . . 4 (𝐴 ∈ V β†’ ((rankβ€˜π΄) = 1o β†’ 𝐴 = 1o))
726, 71mpcom 38 . . 3 ((rankβ€˜π΄) = 1o β†’ 𝐴 = 1o)
73 fveq2 6885 . . . 4 (𝐴 = 1o β†’ (rankβ€˜π΄) = (rankβ€˜1o))
74 r111 9772 . . . . . . 7 𝑅1:On–1-1β†’V
75 f1dm 6785 . . . . . . 7 (𝑅1:On–1-1β†’V β†’ dom 𝑅1 = On)
7674, 75ax-mp 5 . . . . . 6 dom 𝑅1 = On
7749, 76eleqtrri 2826 . . . . 5 1o ∈ dom 𝑅1
78 rankonid 9826 . . . . 5 (1o ∈ dom 𝑅1 ↔ (rankβ€˜1o) = 1o)
7977, 78mpbi 229 . . . 4 (rankβ€˜1o) = 1o
8073, 79eqtrdi 2782 . . 3 (𝐴 = 1o β†’ (rankβ€˜π΄) = 1o)
8172, 80impbii 208 . 2 ((rankβ€˜π΄) = 1o ↔ 𝐴 = 1o)
8261eqeq2i 2739 . 2 (𝐴 = 1o ↔ 𝐴 = {βˆ…})
8381, 82bitri 275 1 ((rankβ€˜π΄) = 1o ↔ 𝐴 = {βˆ…})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  {crab 3426  Vcvv 3468   βŠ† wss 3943  βˆ…c0 4317  π’« cpw 4597  {csn 4623  {cpr 4625  βˆ© cint 4943  dom cdm 5669  Oncon0 6358  suc csuc 6360  β€“1-1β†’wf1 6534  β€˜cfv 6537  1oc1o 8460  π‘…1cr1 9759  rankcrnk 9760
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-reg 9589  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-r1 9761  df-rank 9762
This theorem is referenced by: (None)
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