Theorem nelscottrankgt 35619

Description: If a member of the input set is not a member of the Scott's trick set, then its rank is greater than the rank of a member of the Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)

Assertion

Ref	Expression
nelscottrankgt	⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐴) ∈ (rank‘𝐶))

Proof of Theorem nelscottrankgt

Step	Hyp	Ref	Expression
1		elscottrankss 35617	. . 3 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵) → (rank‘𝐴) ⊆ (rank‘𝐶))
2	1	3adant3 1150	. 2 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐴) ⊆ (rank‘𝐶))
3		scottrankeqel 35618	. . . . . 6 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (rank‘𝐶) = (rank‘𝐴)) → 𝐶 ∈ Scott 𝐵)
4	3	3expia 1139	. . . . 5 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵) → ((rank‘𝐶) = (rank‘𝐴) → 𝐶 ∈ Scott 𝐵))
5	4	necon3bd 2971	. . . 4 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵) → (¬ 𝐶 ∈ Scott 𝐵 → (rank‘𝐶) ≠ (rank‘𝐴)))
6	5	3impia 1135	. . 3 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐶) ≠ (rank‘𝐴))
7	6	necomd 3012	. 2 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐴) ≠ (rank‘𝐶))
8		rankon 9780	. . 3 ⊢ (rank‘𝐴) ∈ On
9		rankon 9780	. . 3 ⊢ (rank‘𝐶) ∈ On
10		onelpss 6402	. . 3 ⊢ (((rank‘𝐴) ∈ On ∧ (rank‘𝐶) ∈ On) → ((rank‘𝐴) ∈ (rank‘𝐶) ↔ ((rank‘𝐴) ⊆ (rank‘𝐶) ∧ (rank‘𝐴) ≠ (rank‘𝐶))))
11	8, 9, 10	mp2an 705	. 2 ⊢ ((rank‘𝐴) ∈ (rank‘𝐶) ↔ ((rank‘𝐴) ⊆ (rank‘𝐶) ∧ (rank‘𝐴) ≠ (rank‘𝐶)))
12	2, 7, 11	sylanbrc 595	1 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐴) ∈ (rank‘𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957   ⊆ wss 3902  Oncon0 6361  ‘cfv 6537  rankcrnk 9748  Scott cscott 9870

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-reg 9567  ax-inf2 9623

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-r1 9749  df-rank 9750  df-scott 9871

This theorem is used by: (None)