Theorem nosepssdm 27667

Description: Given two non-equal surreals, their separator is less-than or equal to the domain of one of them. Part of Lemma 2.1.1 of [Lipparini] p. 3. (Contributed by Scott Fenton, 6-Dec-2021.)

Assertion

Ref	Expression
nosepssdm	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ⊆ dom 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nosepssdm

Step	Hyp	Ref	Expression
1		nosepne 27661	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}))
2	1	neneqd 2938	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ¬ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}))
3		nodmord 27634	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
4	3	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → Ord dom 𝐴)
5		ordn2lp 6338	. . . . . . . 8 ⊢ (Ord dom 𝐴 → ¬ (dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴))
6	4, 5	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ¬ (dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴))
7		imnan 399	. . . . . . 7 ⊢ ((dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} → ¬ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴) ↔ ¬ (dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴))
8	6, 7	sylibr 234	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → (dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} → ¬ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴))
9	8	imp 406	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ¬ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴)
10		ndmfv 6867	. . . . 5 ⊢ (¬ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐴 → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅)
11	9, 10	syl 17	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅)
12		nosepeq 27666	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘dom 𝐴) = (𝐵‘dom 𝐴))
13		simpl1 1193	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → 𝐴 ∈ No )
14		ordirr 6336	. . . . . . . . . 10 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
15		ndmfv 6867	. . . . . . . . . 10 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
16	13, 3, 14, 15	4syl 19	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘dom 𝐴) = ∅)
17	16	eqeq1d 2739	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) ↔ ∅ = (𝐵‘dom 𝐴)))
18		eqcom 2744	. . . . . . . 8 ⊢ (∅ = (𝐵‘dom 𝐴) ↔ (𝐵‘dom 𝐴) = ∅)
19	17, 18	bitrdi 287	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) ↔ (𝐵‘dom 𝐴) = ∅))
20		simpl2 1194	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → 𝐵 ∈ No )
21		nofun 27630	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → Fun 𝐵)
22	20, 21	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → Fun 𝐵)
23		nosgnn0 27639	. . . . . . . . . . 11 ⊢ ¬ ∅ ∈ {1o, 2o}
24		norn 27632	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ No → ran 𝐵 ⊆ {1o, 2o})
25	20, 24	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ran 𝐵 ⊆ {1o, 2o})
26	25	sseld 3921	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (∅ ∈ ran 𝐵 → ∅ ∈ {1o, 2o}))
27	23, 26	mtoi 199	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ¬ ∅ ∈ ran 𝐵)
28		funeldmb 7308	. . . . . . . . . 10 ⊢ ((Fun 𝐵 ∧ ¬ ∅ ∈ ran 𝐵) → (dom 𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅))
29	22, 27, 28	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (dom 𝐴 ∈ dom 𝐵 ↔ (𝐵‘dom 𝐴) ≠ ∅))
30	29	necon2bbid 2976	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐵‘dom 𝐴) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐵))
31		nodmord 27634	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ No → Ord dom 𝐵)
32	31	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → Ord dom 𝐵)
33		ordtr1 6362	. . . . . . . . . . 11 ⊢ (Ord dom 𝐵 → ((dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵) → dom 𝐴 ∈ dom 𝐵))
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ((dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵) → dom 𝐴 ∈ dom 𝐵))
35	34	expdimp 452	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵 → dom 𝐴 ∈ dom 𝐵))
36	35	con3d 152	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (¬ dom 𝐴 ∈ dom 𝐵 → ¬ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵))
37	30, 36	sylbid 240	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐵‘dom 𝐴) = ∅ → ¬ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵))
38	19, 37	sylbid 240	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ((𝐴‘dom 𝐴) = (𝐵‘dom 𝐴) → ¬ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵))
39	12, 38	mpd 15	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → ¬ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵)
40		ndmfv 6867	. . . . 5 ⊢ (¬ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ dom 𝐵 → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅)
41	39, 40	syl 17	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = ∅)
42	11, 41	eqtr4d 2775	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}))
43	2, 42	mtand 816	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ¬ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})
44		nosepon 27646	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On)
45		nodmon 27631	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
46	45	3ad2ant1 1134	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → dom 𝐴 ∈ On)
47		ontri1 6352	. . 3 ⊢ ((∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ On ∧ dom 𝐴 ∈ On) → (∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ⊆ dom 𝐴 ↔ ¬ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}))
48	44, 46, 47	syl2anc 585	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → (∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ⊆ dom 𝐴 ↔ ¬ dom 𝐴 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}))
49	43, 48	mpbird 257	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ⊆ dom 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3390   ⊆ wss 3890  ∅c0 4274  {cpr 4570  ∩ cint 4890  dom cdm 5625  ran crn 5626  Ord word 6317  Oncon0 6318  Fun wfun 6487  ‘cfv 6493  1_oc1o 8392  2_oc2o 8393   No csur 27620

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 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-1o 8399  df-2o 8400  df-no 27623  df-lts 27624

This theorem is referenced by:  nosupbnd2lem1  27696  noinfbnd2lem1  27711  noetasuplem4  27717  noetainflem4  27721