Theorem nosep2o 33482

Description: If the value of a surreal at a separator is 2_o then the surreal is greater. (Contributed by Scott Fenton, 7-Dec-2021.)

Assertion

Ref	Expression
nosep2o	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → 𝐵 <s 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nosep2o

Step	Hyp	Ref	Expression
1		simp2 1134	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ No )
2		simp1 1133	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ No )
3		simp3 1135	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
4	3	necomd 3006	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → 𝐵 ≠ 𝐴)
5		nosepne 33480	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ 𝐵 ≠ 𝐴) → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ≠ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}))
6	1, 2, 4, 5	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ≠ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}))
7	6	adantr 484	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ≠ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}))
8		simpr 488	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o)
9	7, 8	neeqtrd 3020	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ≠ 2o)
10	9	neneqd 2956	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → ¬ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o)
11		simpl2 1189	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → 𝐵 ∈ No )
12		nofv 33457	. . . . . . . . 9 ⊢ (𝐵 ∈ No → ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o))
13	11, 12	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o))
14		3orel3 33185	. . . . . . . 8 ⊢ (¬ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o → (((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o)))
15	10, 13, 14	sylc 65	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o))
16	15	orcomd 868	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅))
17	16, 8	jca 515	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → (((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o))
18		andir 1006	. . . . 5 ⊢ ((((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∨ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) ↔ (((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o)))
19	17, 18	sylib 221	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → (((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o)))
20		3mix2 1328	. . . . 5 ⊢ (((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → (((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o)))
21		3mix3 1329	. . . . 5 ⊢ (((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → (((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o)))
22	20, 21	jaoi 854	. . . 4 ⊢ ((((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o)) → (((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o)))
23	19, 22	syl 17	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → (((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o)))
24		fvex 6676	. . . 4 ⊢ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ∈ V
25		fvex 6676	. . . 4 ⊢ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ∈ V
26	24, 25	brtp 33244	. . 3 ⊢ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ↔ (((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 1o ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) ∨ ((𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = ∅ ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o)))
27	23, 26	sylibr 237	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}))
28		simpl1 1188	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → 𝐴 ∈ No )
29		sltval2 33456	. . 3 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (𝐵 <s 𝐴 ↔ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)})))
30	11, 28, 29	syl2anc 587	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → (𝐵 <s 𝐴 ↔ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)})))
31	27, 30	mpbird 260	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → 𝐵 <s 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  {crab 3074  ∅c0 4227  {ctp 4529  ⟨cop 4531  ∩ cint 4841   class class class wbr 5036  Oncon0 6174  ‘cfv 6340  1_oc1o 8111  2_oc2o 8112   No csur 33440   <s cslt 33441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6177  df-on 6178  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-1o 8118  df-2o 8119  df-no 33443  df-slt 33444

This theorem is referenced by:  noetainflem4  33540