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Theorem scottrankd 42439
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 42436 . . . 4 Scott 𝐴 ∈ V
21rankval4 9761 . . 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 481 . . . . . 6 ((𝜑𝑥 ∈ Scott 𝐴) → 𝐵 ∈ Scott 𝐴)
6 simpr 485 . . . . . 6 ((𝜑𝑥 ∈ Scott 𝐴) → 𝑥 ∈ Scott 𝐴)
75, 6scottelrankd 42438 . . . . 5 ((𝜑𝑥 ∈ Scott 𝐴) → (rank‘𝐵) ⊆ (rank‘𝑥))
86, 5scottelrankd 42438 . . . . 5 ((𝜑𝑥 ∈ Scott 𝐴) → (rank‘𝑥) ⊆ (rank‘𝐵))
97, 8eqssd 3959 . . . 4 ((𝜑𝑥 ∈ Scott 𝐴) → (rank‘𝐵) = (rank‘𝑥))
109suceqd 42395 . . 3 ((𝜑𝑥 ∈ Scott 𝐴) → suc (rank‘𝐵) = suc (rank‘𝑥))
1110iuneq2dv 4976 . 2 (𝜑 𝑥 ∈ Scott 𝐴 suc (rank‘𝐵) = 𝑥 ∈ Scott 𝐴 suc (rank‘𝑥))
124ne0d 4293 . . 3 (𝜑 → Scott 𝐴 ≠ ∅)
13 iunconst 4961 . . 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 396   = wceq 1541  wcel 2106  wne 2941  c0 4280   ciun 4952  suc csuc 6317  cfv 6493  rankcrnk 9657  Scott cscott 42426
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-reg 9486  ax-inf2 9535
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-om 7795  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-r1 9658  df-rank 9659  df-scott 42427
This theorem is referenced by:  gruscottcld  42440
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