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Theorem scottrankd 40807
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 40804 . . . 4 Scott 𝐴 ∈ V
21rankval4 9282 . . 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 484 . . . . . 6 ((𝜑𝑥 ∈ Scott 𝐴) → 𝐵 ∈ Scott 𝐴)
6 simpr 488 . . . . . 6 ((𝜑𝑥 ∈ Scott 𝐴) → 𝑥 ∈ Scott 𝐴)
75, 6scottelrankd 40806 . . . . 5 ((𝜑𝑥 ∈ Scott 𝐴) → (rank‘𝐵) ⊆ (rank‘𝑥))
86, 5scottelrankd 40806 . . . . 5 ((𝜑𝑥 ∈ Scott 𝐴) → (rank‘𝑥) ⊆ (rank‘𝐵))
97, 8eqssd 3968 . . . 4 ((𝜑𝑥 ∈ Scott 𝐴) → (rank‘𝐵) = (rank‘𝑥))
109suceqd 40768 . . 3 ((𝜑𝑥 ∈ Scott 𝐴) → suc (rank‘𝐵) = suc (rank‘𝑥))
1110iuneq2dv 4926 . 2 (𝜑 𝑥 ∈ Scott 𝐴 suc (rank‘𝐵) = 𝑥 ∈ Scott 𝐴 suc (rank‘𝑥))
124ne0d 4282 . . 3 (𝜑 → Scott 𝐴 ≠ ∅)
13 iunconst 4911 . . 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 2865 1 (𝜑 → (rank‘Scott 𝐴) = suc (rank‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3013  c0 4274   ciun 4902  suc csuc 6176  cfv 6338  rankcrnk 9178  Scott cscott 40794
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446  ax-reg 9042  ax-inf2 9090
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-int 4860  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-om 7566  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-r1 9179  df-rank 9180  df-scott 40795
This theorem is referenced by:  gruscottcld  40808
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