Theorem nosep2o 27646

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 1138	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ No )
2		simp1 1137	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ No )
3		simp3 1139	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
4	3	necomd 2987	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → 𝐵 ≠ 𝐴)
5		nosepne 27644	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ≠ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}))
7	6	adantr 480	. . . . . . . . . 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 484	. . . . . . . . . 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 3001	. . . . . . . . 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 2937	. . . . . . . 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 1194	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → 𝐵 ∈ No )
12		nofv 27621	. . . . . . . . 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 1489	. . . . . . . 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 872	. . . . . 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 511	. . . . 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 1011	. . . . 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 218	. . . 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 1333	. . . . 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 1334	. . . . 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 858	. . . 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 6853	. . . 4 ⊢ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ∈ V
25		fvex 6853	. . . 4 ⊢ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) ∈ V
26	24, 25	brtp 5478	. . 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 234	. 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 1193	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐵‘𝑥) ≠ (𝐴‘𝑥)}) = 2o) → 𝐴 ∈ No )
29		ltsval2 27620	. . 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 585	. 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 257	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 206   ∧ wa 395   ∨ wo 848   ∨ w3o 1086   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  {crab 3389  ∅c0 4273  {ctp 4571  ⟨cop 4573  ∩ cint 4889   class class class wbr 5085  Oncon0 6323  ‘cfv 6498  1_oc1o 8398  2_oc2o 8399   No csur 27603   <s clts 27604

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607

This theorem is referenced by:  noetainflem4  27704