Theorem noseponlem 27628

Description: Lemma for nosepon 27629. 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 27614	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
2	1	3ad2ant1 1134	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → dom 𝐴 ∈ On)
3		nodmord 27617	. . . . . . 7 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
4		ordirr 6341	. . . . . . 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 6872	. . . . 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 27622	. . . . . . 7 ⊢ ¬ ∅ ∈ {1o, 2o}
10		elno3 27619	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No ↔ (𝐵:dom 𝐵⟶{1o, 2o} ∧ dom 𝐵 ∈ On))
11	10	simplbi 496	. . . . . . . . . 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 7037	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → (𝐵‘dom 𝐴) ∈ {1o, 2o})
15		eleq1 2824	. . . . . . . 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 2939	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → (𝐵‘dom 𝐴) ≠ ∅)
19	18	necomd 2987	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ∅ ≠ (𝐵‘dom 𝐴))
20	8, 19	eqnetrd 2999	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → (𝐴‘dom 𝐴) ≠ (𝐵‘dom 𝐴))
21		fveq2 6840	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴‘𝑥) = (𝐴‘dom 𝐴))
22		fveq2 6840	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐵‘𝑥) = (𝐵‘dom 𝐴))
23	21, 22	neeq12d 2993	. . . 4 ⊢ (𝑥 = dom 𝐴 → ((𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ (𝐴‘dom 𝐴) ≠ (𝐵‘dom 𝐴)))
24	23	rspcev 3564	. . 3 ⊢ ((dom 𝐴 ∈ On ∧ (𝐴‘dom 𝐴) ≠ (𝐵‘dom 𝐴)) → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥))
25	2, 20, 24	syl2anc 585	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥))
26		df-ne 2933	. . . 4 ⊢ ((𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
27	26	rexbii 3084	. . 3 ⊢ (∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ∃𝑥 ∈ On ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
28		rexnal 3089	. . 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 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  ∅c0 4273  {cpr 4569  dom cdm 5631  Ord word 6322  Oncon0 6323  ⟶wf 6494  ‘cfv 6498  1_oc1o 8398  2_oc2o 8399   No csur 27603

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-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-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-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  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-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

This theorem is referenced by:  nosepon  27629