Theorem nelscottrankgt 35484

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 35482	. . 3 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵) → (rank‘𝐴) ⊆ (rank‘𝐶))
2	1	3adant3 1148	. 2 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐴) ⊆ (rank‘𝐶))
3		scottrankeqel 35483	. . . . . 6 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (rank‘𝐶) = (rank‘𝐴)) → 𝐶 ∈ Scott 𝐵)
4	3	3expia 1137	. . . . 5 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵) → ((rank‘𝐶) = (rank‘𝐴) → 𝐶 ∈ Scott 𝐵))
5	4	necon3bd 2970	. . . 4 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵) → (¬ 𝐶 ∈ Scott 𝐵 → (rank‘𝐶) ≠ (rank‘𝐴)))
6	5	3impia 1133	. . 3 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐶) ≠ (rank‘𝐴))
7	6	necomd 3011	. 2 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐴) ≠ (rank‘𝐶))
8		rankon 9766	. . 3 ⊢ (rank‘𝐴) ∈ On
9		rankon 9766	. . 3 ⊢ (rank‘𝐶) ∈ On
10		onelpss 6401	. . 3 ⊢ (((rank‘𝐴) ∈ On ∧ (rank‘𝐶) ∈ On) → ((rank‘𝐴) ∈ (rank‘𝐶) ↔ ((rank‘𝐴) ⊆ (rank‘𝐶) ∧ (rank‘𝐴) ≠ (rank‘𝐶))))
11	8, 9, 10	mp2an 704	. 2 ⊢ ((rank‘𝐴) ∈ (rank‘𝐶) ↔ ((rank‘𝐴) ⊆ (rank‘𝐶) ∧ (rank‘𝐴) ≠ (rank‘𝐶)))
12	2, 7, 11	sylanbrc 594	1 ⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐴) ∈ (rank‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ≠ wne 2956   ⊆ wss 3904  Oncon0 6360  ‘cfv 6536  rankcrnk 9734  Scott cscott 9856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-reg 9553  ax-inf2 9609

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-r1 9735  df-rank 9736  df-scott 9857

This theorem is referenced by: (None)