Theorem nelscottrankgt 35679

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 35677	. . 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 35678	. . . . . 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 2969	. . . 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 3010	. 2 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐴) ≠ (rank‘𝐶))
8		rankon 9777	. . 3 ⊢ (rank‘𝐴) ∈ On
9		rankon 9777	. . 3 ⊢ (rank‘𝐶) ∈ On
10		onelpss 6392	. . 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 2955   ⊆ wss 3898  Oncon0 6351  ‘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: (None)