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Theorem rankeq1o 34756
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 8434 . . . . . . 7 1o β‰  βˆ…
2 neeq1 3006 . . . . . . 7 ((rankβ€˜π΄) = 1o β†’ ((rankβ€˜π΄) β‰  βˆ… ↔ 1o β‰  βˆ…))
31, 2mpbiri 257 . . . . . 6 ((rankβ€˜π΄) = 1o β†’ (rankβ€˜π΄) β‰  βˆ…)
43neneqd 2948 . . . . 5 ((rankβ€˜π΄) = 1o β†’ Β¬ (rankβ€˜π΄) = βˆ…)
5 fvprc 6834 . . . . 5 (Β¬ 𝐴 ∈ V β†’ (rankβ€˜π΄) = βˆ…)
64, 5nsyl2 141 . . . 4 ((rankβ€˜π΄) = 1o β†’ 𝐴 ∈ V)
7 fveqeq2 6851 . . . . . 6 (π‘₯ = 𝐴 β†’ ((rankβ€˜π‘₯) = 1o ↔ (rankβ€˜π΄) = 1o))
8 eqeq1 2740 . . . . . 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 3006 . . . . . . . 8 ((rankβ€˜π‘₯) = 1o β†’ ((rankβ€˜π‘₯) β‰  βˆ… ↔ 1o β‰  βˆ…))
111, 10mpbiri 257 . . . . . . 7 ((rankβ€˜π‘₯) = 1o β†’ (rankβ€˜π‘₯) β‰  βˆ…)
12 vex 3449 . . . . . . . . 9 π‘₯ ∈ V
1312rankeq0 9797 . . . . . . . 8 (π‘₯ = βˆ… ↔ (rankβ€˜π‘₯) = βˆ…)
1413necon3bii 2996 . . . . . . 7 (π‘₯ β‰  βˆ… ↔ (rankβ€˜π‘₯) β‰  βˆ…)
1511, 14sylibr 233 . . . . . 6 ((rankβ€˜π‘₯) = 1o β†’ π‘₯ β‰  βˆ…)
1612rankval 9752 . . . . . . . 8 (rankβ€˜π‘₯) = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)}
1716eqeq1i 2741 . . . . . . 7 ((rankβ€˜π‘₯) = 1o ↔ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o)
18 ssrab2 4037 . . . . . . . . . . 11 {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} βŠ† On
19 elirr 9533 . . . . . . . . . . . . . 14 Β¬ 1o ∈ 1o
20 1oex 8422 . . . . . . . . . . . . . . 15 1o ∈ V
21 id 22 . . . . . . . . . . . . . . 15 (V = 1o β†’ V = 1o)
2220, 21eleqtrid 2843 . . . . . . . . . . . . . 14 (V = 1o β†’ 1o ∈ 1o)
2319, 22mto 196 . . . . . . . . . . . . 13 Β¬ V = 1o
24 inteq 4910 . . . . . . . . . . . . . . 15 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = βˆ… β†’ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = ∩ βˆ…)
25 int0 4923 . . . . . . . . . . . . . . 15 ∩ βˆ… = V
2624, 25eqtrdi 2792 . . . . . . . . . . . . . 14 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = βˆ… β†’ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = V)
2726eqeq1d 2738 . . . . . . . . . . . . 13 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = βˆ… β†’ (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o ↔ V = 1o))
2823, 27mtbiri 326 . . . . . . . . . . . 12 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = βˆ… β†’ Β¬ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o)
2928necon2ai 2973 . . . . . . . . . . 11 (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o β†’ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} β‰  βˆ…)
30 onint 7725 . . . . . . . . . . 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 587 . . . . . . . . . 10 (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} = 1o β†’ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ∈ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)})
32 eleq1 2825 . . . . . . . . . 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 6383 . . . . . . . . . . . . 13 (𝑦 = 1o β†’ suc 𝑦 = suc 1o)
3534fveq2d 6846 . . . . . . . . . . . 12 (𝑦 = 1o β†’ (𝑅1β€˜suc 𝑦) = (𝑅1β€˜suc 1o))
36 df-1o 8412 . . . . . . . . . . . . . . . . 17 1o = suc βˆ…
3736fveq2i 6845 . . . . . . . . . . . . . . . 16 (𝑅1β€˜1o) = (𝑅1β€˜suc βˆ…)
38 0elon 6371 . . . . . . . . . . . . . . . . 17 βˆ… ∈ On
39 r1suc 9706 . . . . . . . . . . . . . . . . 17 (βˆ… ∈ On β†’ (𝑅1β€˜suc βˆ…) = 𝒫 (𝑅1β€˜βˆ…))
4038, 39ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑅1β€˜suc βˆ…) = 𝒫 (𝑅1β€˜βˆ…)
41 r10 9704 . . . . . . . . . . . . . . . . 17 (𝑅1β€˜βˆ…) = βˆ…
4241pweqi 4576 . . . . . . . . . . . . . . . 16 𝒫 (𝑅1β€˜βˆ…) = 𝒫 βˆ…
4337, 40, 423eqtri 2768 . . . . . . . . . . . . . . 15 (𝑅1β€˜1o) = 𝒫 βˆ…
4443pweqi 4576 . . . . . . . . . . . . . 14 𝒫 (𝑅1β€˜1o) = 𝒫 𝒫 βˆ…
45 pw0 4772 . . . . . . . . . . . . . . 15 𝒫 βˆ… = {βˆ…}
4645pweqi 4576 . . . . . . . . . . . . . 14 𝒫 𝒫 βˆ… = 𝒫 {βˆ…}
47 pwpw0 4773 . . . . . . . . . . . . . 14 𝒫 {βˆ…} = {βˆ…, {βˆ…}}
4844, 46, 473eqtrri 2769 . . . . . . . . . . . . 13 {βˆ…, {βˆ…}} = 𝒫 (𝑅1β€˜1o)
49 1on 8424 . . . . . . . . . . . . . 14 1o ∈ On
50 r1suc 9706 . . . . . . . . . . . . . 14 (1o ∈ On β†’ (𝑅1β€˜suc 1o) = 𝒫 (𝑅1β€˜1o))
5149, 50ax-mp 5 . . . . . . . . . . . . 13 (𝑅1β€˜suc 1o) = 𝒫 (𝑅1β€˜1o)
5248, 51eqtr4i 2767 . . . . . . . . . . . 12 {βˆ…, {βˆ…}} = (𝑅1β€˜suc 1o)
5335, 52eqtr4di 2794 . . . . . . . . . . 11 (𝑦 = 1o β†’ (𝑅1β€˜suc 𝑦) = {βˆ…, {βˆ…}})
5453eleq2d 2823 . . . . . . . . . 10 (𝑦 = 1o β†’ (π‘₯ ∈ (𝑅1β€˜suc 𝑦) ↔ π‘₯ ∈ {βˆ…, {βˆ…}}))
5554elrab 3645 . . . . . . . . 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 4609 . . . . . . . . . 10 (π‘₯ ∈ {βˆ…, {βˆ…}} ↔ (π‘₯ = βˆ… ∨ π‘₯ = {βˆ…}))
58 df-ne 2944 . . . . . . . . . . . 12 (π‘₯ β‰  βˆ… ↔ Β¬ π‘₯ = βˆ…)
59 orel1 887 . . . . . . . . . . . 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 8419 . . . . . . . . . . . . 13 1o = {βˆ…}
62 eqeq2 2748 . . . . . . . . . . . . 13 (π‘₯ = {βˆ…} β†’ (1o = π‘₯ ↔ 1o = {βˆ…}))
6361, 62mpbiri 257 . . . . . . . . . . . 12 (π‘₯ = {βˆ…} β†’ 1o = π‘₯)
6463eqcomd 2742 . . . . . . . . . . 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 482 . . . . . . . 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 3525 . . . 4 (𝐴 ∈ V β†’ ((rankβ€˜π΄) = 1o β†’ 𝐴 = 1o))
726, 71mpcom 38 . . 3 ((rankβ€˜π΄) = 1o β†’ 𝐴 = 1o)
73 fveq2 6842 . . . 4 (𝐴 = 1o β†’ (rankβ€˜π΄) = (rankβ€˜1o))
74 r111 9711 . . . . . . 7 𝑅1:On–1-1β†’V
75 f1dm 6742 . . . . . . 7 (𝑅1:On–1-1β†’V β†’ dom 𝑅1 = On)
7674, 75ax-mp 5 . . . . . 6 dom 𝑅1 = On
7749, 76eleqtrri 2836 . . . . 5 1o ∈ dom 𝑅1
78 rankonid 9765 . . . . 5 (1o ∈ dom 𝑅1 ↔ (rankβ€˜1o) = 1o)
7977, 78mpbi 229 . . . 4 (rankβ€˜1o) = 1o
8073, 79eqtrdi 2792 . . 3 (𝐴 = 1o β†’ (rankβ€˜π΄) = 1o)
8172, 80impbii 208 . 2 ((rankβ€˜π΄) = 1o ↔ 𝐴 = 1o)
8261eqeq2i 2749 . 2 (𝐴 = 1o ↔ 𝐴 = {βˆ…})
8381, 82bitri 274 1 ((rankβ€˜π΄) = 1o ↔ 𝐴 = {βˆ…})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   β‰  wne 2943  {crab 3407  Vcvv 3445   βŠ† wss 3910  βˆ…c0 4282  π’« cpw 4560  {csn 4586  {cpr 4588  βˆ© cint 4907  dom cdm 5633  Oncon0 6317  suc csuc 6319  β€“1-1β†’wf1 6493  β€˜cfv 6496  1oc1o 8405  π‘…1cr1 9698  rankcrnk 9699
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-r1 9700  df-rank 9701
This theorem is referenced by: (None)
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