Theorem scottrankeqel 35790

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 35787	. . . 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 35786	. . . 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 1569   ∈ wcel 2144  ∩ cint 4906   “ cima 5653  ‘cfv 6538  rankcrnk 9767  Scott cscott 9928

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9586  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  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 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768  df-rank 9769  df-scott 9929

This theorem is used by:  nelscottrankgt  35791