Users' Mathboxes Mathbox for Rohan Ridenour < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  scottrankd Structured version   Visualization version   GIF version

Theorem scottrankd 43750
Description: Rank of a nonempty Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypothesis
Ref Expression
scottrankd.1 (πœ‘ β†’ 𝐡 ∈ Scott 𝐴)
Assertion
Ref Expression
scottrankd (πœ‘ β†’ (rankβ€˜Scott 𝐴) = suc (rankβ€˜π΅))

Proof of Theorem scottrankd
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 scottex2 43747 . . . 4 Scott 𝐴 ∈ V
21rankval4 9890 . . 3 (rankβ€˜Scott 𝐴) = βˆͺ π‘₯ ∈ Scott 𝐴 suc (rankβ€˜π‘₯)
32a1i 11 . 2 (πœ‘ β†’ (rankβ€˜Scott 𝐴) = βˆͺ π‘₯ ∈ Scott 𝐴 suc (rankβ€˜π‘₯))
4 scottrankd.1 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ Scott 𝐴)
54adantr 479 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ Scott 𝐴) β†’ 𝐡 ∈ Scott 𝐴)
6 simpr 483 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ Scott 𝐴) β†’ π‘₯ ∈ Scott 𝐴)
75, 6scottelrankd 43749 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ Scott 𝐴) β†’ (rankβ€˜π΅) βŠ† (rankβ€˜π‘₯))
86, 5scottelrankd 43749 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ Scott 𝐴) β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π΅))
97, 8eqssd 3990 . . . 4 ((πœ‘ ∧ π‘₯ ∈ Scott 𝐴) β†’ (rankβ€˜π΅) = (rankβ€˜π‘₯))
109suceqd 43706 . . 3 ((πœ‘ ∧ π‘₯ ∈ Scott 𝐴) β†’ suc (rankβ€˜π΅) = suc (rankβ€˜π‘₯))
1110iuneq2dv 5015 . 2 (πœ‘ β†’ βˆͺ π‘₯ ∈ Scott 𝐴 suc (rankβ€˜π΅) = βˆͺ π‘₯ ∈ Scott 𝐴 suc (rankβ€˜π‘₯))
124ne0d 4331 . . 3 (πœ‘ β†’ Scott 𝐴 β‰  βˆ…)
13 iunconst 5000 . . 3 (Scott 𝐴 β‰  βˆ… β†’ βˆͺ π‘₯ ∈ Scott 𝐴 suc (rankβ€˜π΅) = suc (rankβ€˜π΅))
1412, 13syl 17 . 2 (πœ‘ β†’ βˆͺ π‘₯ ∈ Scott 𝐴 suc (rankβ€˜π΅) = suc (rankβ€˜π΅))
153, 11, 143eqtr2d 2771 1 (πœ‘ β†’ (rankβ€˜Scott 𝐴) = suc (rankβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ…c0 4318  βˆͺ ciun 4991  suc csuc 6366  β€˜cfv 6543  rankcrnk 9786  Scott cscott 43737
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-reg 9615  ax-inf2 9664
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-om 7869  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-r1 9787  df-rank 9788  df-scott 43738
This theorem is referenced by:  gruscottcld  43751
  Copyright terms: Public domain W3C validator