Theorem dfscott2 35672

Description: Alternate definition of a Scott's trick set. (Contributed by BTernaryTau, 8-Jul-2026.)

Assertion

Ref	Expression
dfscott2	⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) = ∩ (rank “ 𝐴)}

Distinct variable group:   𝑥,𝐴

Proof of Theorem dfscott2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-scott 9900	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
2		rankfn 35667	. . . 4 ⊢ rank Fn V
3		ssv 3954	. . . 4 ⊢ 𝐴 ⊆ V
4		fnfvintima 35646	. . . 4 ⊢ ((rank Fn V ∧ 𝐴 ⊆ V ∧ 𝑥 ∈ 𝐴) → ((rank‘𝑥) = ∩ (rank “ 𝐴) ↔ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)))
5	2, 3, 4	mp3an12 1480	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((rank‘𝑥) = ∩ (rank “ 𝐴) ↔ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)))
6	5	rabbiia 3416	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) = ∩ (rank “ 𝐴)} = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
7	1, 6	eqtr4i 2786	1 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) = ∩ (rank “ 𝐴)}

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3898  ∩ cint 4906   “ cima 5650   Fn wfn 6522  ‘cfv 6527  rankcrnk 9745  Scott cscott 9899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-reg 9564  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-r1 9746  df-rank 9747  df-scott 9900

This theorem is used by:  dfscott3  35673  elscott2  35674  elscottrank  35675