Theorem scottrankeqel 35678

Description: If a member of the input set has the same rank as a member of the Scott's trick set, then it is also a member of the Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)

Assertion

Ref	Expression
scottrankeqel	⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (rank‘𝐶) = (rank‘𝐴)) → 𝐶 ∈ Scott 𝐵)

Proof of Theorem scottrankeqel

Step	Hyp	Ref	Expression
1		elscottrank 35675	. . . 4 ⊢ (𝐴 ∈ Scott 𝐵 → (rank‘𝐴) = ∩ (rank “ 𝐵))
2	1	3ad2ant1 1151	. . 3 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (rank‘𝐶) = (rank‘𝐴)) → (rank‘𝐴) = ∩ (rank “ 𝐵))
3		eqtr 2780	. . . 4 ⊢ (((rank‘𝐶) = (rank‘𝐴) ∧ (rank‘𝐴) = ∩ (rank “ 𝐵)) → (rank‘𝐶) = ∩ (rank “ 𝐵))
4	3	3ad2antl3 1206	. . 3 ⊢ (((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (rank‘𝐶) = (rank‘𝐴)) ∧ (rank‘𝐴) = ∩ (rank “ 𝐵)) → (rank‘𝐶) = ∩ (rank “ 𝐵))
5	2, 4	mpdan 700	. 2 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (rank‘𝐶) = (rank‘𝐴)) → (rank‘𝐶) = ∩ (rank “ 𝐵))
6		elscott2 35674	. . . 4 ⊢ (𝐶 ∈ Scott 𝐵 ↔ (𝐶 ∈ 𝐵 ∧ (rank‘𝐶) = ∩ (rank “ 𝐵)))
7	6	baib 545	. . 3 ⊢ (𝐶 ∈ 𝐵 → (𝐶 ∈ Scott 𝐵 ↔ (rank‘𝐶) = ∩ (rank “ 𝐵)))
8	7	3ad2ant2 1152	. 2 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (rank‘𝐶) = (rank‘𝐴)) → (𝐶 ∈ Scott 𝐵 ↔ (rank‘𝐶) = ∩ (rank “ 𝐵)))
9	5, 8	mpbird 260	1 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (rank‘𝐶) = (rank‘𝐴)) → 𝐶 ∈ Scott 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∩ cint 4906   “ cima 5650  ‘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:  nelscottrankgt  35679