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Theorem scottrankd 44267
Description: Rank of a nonempty Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypothesis
Ref Expression
scottrankd.1 (𝜑𝐵 ∈ Scott 𝐴)
Assertion
Ref Expression
scottrankd (𝜑 → (rank‘Scott 𝐴) = suc (rank‘𝐵))

Proof of Theorem scottrankd
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 scottex2 44264 . . . 4 Scott 𝐴 ∈ V
21rankval4 9907 . . 3 (rank‘Scott 𝐴) = 𝑥 ∈ Scott 𝐴 suc (rank‘𝑥)
32a1i 11 . 2 (𝜑 → (rank‘Scott 𝐴) = 𝑥 ∈ Scott 𝐴 suc (rank‘𝑥))
4 scottrankd.1 . . . . . . 7 (𝜑𝐵 ∈ Scott 𝐴)
54adantr 480 . . . . . 6 ((𝜑𝑥 ∈ Scott 𝐴) → 𝐵 ∈ Scott 𝐴)
6 simpr 484 . . . . . 6 ((𝜑𝑥 ∈ Scott 𝐴) → 𝑥 ∈ Scott 𝐴)
75, 6scottelrankd 44266 . . . . 5 ((𝜑𝑥 ∈ Scott 𝐴) → (rank‘𝐵) ⊆ (rank‘𝑥))
86, 5scottelrankd 44266 . . . . 5 ((𝜑𝑥 ∈ Scott 𝐴) → (rank‘𝑥) ⊆ (rank‘𝐵))
97, 8eqssd 4001 . . . 4 ((𝜑𝑥 ∈ Scott 𝐴) → (rank‘𝐵) = (rank‘𝑥))
109suceqd 44223 . . 3 ((𝜑𝑥 ∈ Scott 𝐴) → suc (rank‘𝐵) = suc (rank‘𝑥))
1110iuneq2dv 5016 . 2 (𝜑 𝑥 ∈ Scott 𝐴 suc (rank‘𝐵) = 𝑥 ∈ Scott 𝐴 suc (rank‘𝑥))
124ne0d 4342 . . 3 (𝜑 → Scott 𝐴 ≠ ∅)
13 iunconst 5001 . . 3 (Scott 𝐴 ≠ ∅ → 𝑥 ∈ Scott 𝐴 suc (rank‘𝐵) = suc (rank‘𝐵))
1412, 13syl 17 . 2 (𝜑 𝑥 ∈ Scott 𝐴 suc (rank‘𝐵) = suc (rank‘𝐵))
153, 11, 143eqtr2d 2783 1 (𝜑 → (rank‘Scott 𝐴) = suc (rank‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  c0 4333   ciun 4991  suc csuc 6386  cfv 6561  rankcrnk 9803  Scott cscott 44254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805  df-scott 44255
This theorem is referenced by:  gruscottcld  44268
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