Theorem noseponlem 27709

Description: Lemma for nosepon 27710. Consider a case of proper subset domain. (Contributed by Scott Fenton, 21-Sep-2020.)

Assertion

Ref	Expression
noseponlem	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem noseponlem

Step	Hyp	Ref	Expression
1		nodmon 27695	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
2	1	3ad2ant1 1134	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → dom 𝐴 ∈ On)
3		nodmord 27698	. . . . . . 7 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
4		ordirr 6402	. . . . . . 7 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
6	5	3ad2ant1 1134	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ dom 𝐴 ∈ dom 𝐴)
7		ndmfv 6941	. . . . 5 ⊢ (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
8	6, 7	syl 17	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → (𝐴‘dom 𝐴) = ∅)
9		nosgnn0 27703	. . . . . . 7 ⊢ ¬ ∅ ∈ {1o, 2o}
10		elno3 27700	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No ↔ (𝐵:dom 𝐵⟶{1o, 2o} ∧ dom 𝐵 ∈ On))
11	10	simplbi 497	. . . . . . . . . 10 ⊢ (𝐵 ∈ No → 𝐵:dom 𝐵⟶{1o, 2o})
12	11	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → 𝐵:dom 𝐵⟶{1o, 2o})
13		simp3 1139	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → dom 𝐴 ∈ dom 𝐵)
14	12, 13	ffvelcdmd 7105	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → (𝐵‘dom 𝐴) ∈ {1o, 2o})
15		eleq1 2829	. . . . . . . 8 ⊢ ((𝐵‘dom 𝐴) = ∅ → ((𝐵‘dom 𝐴) ∈ {1o, 2o} ↔ ∅ ∈ {1o, 2o}))
16	14, 15	syl5ibcom 245	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ((𝐵‘dom 𝐴) = ∅ → ∅ ∈ {1o, 2o}))
17	9, 16	mtoi 199	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ (𝐵‘dom 𝐴) = ∅)
18	17	neqned 2947	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → (𝐵‘dom 𝐴) ≠ ∅)
19	18	necomd 2996	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ∅ ≠ (𝐵‘dom 𝐴))
20	8, 19	eqnetrd 3008	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → (𝐴‘dom 𝐴) ≠ (𝐵‘dom 𝐴))
21		fveq2 6906	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴‘𝑥) = (𝐴‘dom 𝐴))
22		fveq2 6906	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐵‘𝑥) = (𝐵‘dom 𝐴))
23	21, 22	neeq12d 3002	. . . 4 ⊢ (𝑥 = dom 𝐴 → ((𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ (𝐴‘dom 𝐴) ≠ (𝐵‘dom 𝐴)))
24	23	rspcev 3622	. . 3 ⊢ ((dom 𝐴 ∈ On ∧ (𝐴‘dom 𝐴) ≠ (𝐵‘dom 𝐴)) → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥))
25	2, 20, 24	syl2anc 584	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥))
26		df-ne 2941	. . . 4 ⊢ ((𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
27	26	rexbii 3094	. . 3 ⊢ (∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ∃𝑥 ∈ On ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
28		rexnal 3100	. . 3 ⊢ (∃𝑥 ∈ On ¬ (𝐴‘𝑥) = (𝐵‘𝑥) ↔ ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥))
29	27, 28	bitri 275	. 2 ⊢ (∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥))
30	25, 29	sylib 218	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴‘𝑥) = (𝐵‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  ∅c0 4333  {cpr 4628  dom cdm 5685  Ord word 6383  Oncon0 6384  ⟶wf 6557  ‘cfv 6561  1_oc1o 8499  2_oc2o 8500   No csur 27684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-1o 8506  df-2o 8507  df-no 27687

This theorem is referenced by:  nosepon  27710