Theorem noseponlem 27598

Description: Lemma for nosepon 27599. 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 27584	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
2	1	3ad2ant1 1133	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → dom 𝐴 ∈ On)
3		nodmord 27587	. . . . . . 7 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
4		ordirr 6319	. . . . . . 7 ⊢ (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝐴 ∈ No → ¬ dom 𝐴 ∈ dom 𝐴)
6	5	3ad2ant1 1133	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ dom 𝐴 ∈ dom 𝐴)
7		ndmfv 6849	. . . . 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 27592	. . . . . . 7 ⊢ ¬ ∅ ∈ {1o, 2o}
10		elno3 27589	. . . . . . . . . . 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 1134	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → 𝐵:dom 𝐵⟶{1o, 2o})
13		simp3 1138	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → dom 𝐴 ∈ dom 𝐵)
14	12, 13	ffvelcdmd 7013	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → (𝐵‘dom 𝐴) ∈ {1o, 2o})
15		eleq1 2819	. . . . . . . 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 2935	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → (𝐵‘dom 𝐴) ≠ ∅)
19	18	necomd 2983	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → ∅ ≠ (𝐵‘dom 𝐴))
20	8, 19	eqnetrd 2995	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ dom 𝐴 ∈ dom 𝐵) → (𝐴‘dom 𝐴) ≠ (𝐵‘dom 𝐴))
21		fveq2 6817	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴‘𝑥) = (𝐴‘dom 𝐴))
22		fveq2 6817	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐵‘𝑥) = (𝐵‘dom 𝐴))
23	21, 22	neeq12d 2989	. . . 4 ⊢ (𝑥 = dom 𝐴 → ((𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ (𝐴‘dom 𝐴) ≠ (𝐵‘dom 𝐴)))
24	23	rspcev 3572	. . 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 2929	. . . 4 ⊢ ((𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
27	26	rexbii 3079	. . 3 ⊢ (∃𝑥 ∈ On (𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ∃𝑥 ∈ On ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
28		rexnal 3084	. . 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 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  ∅c0 4278  {cpr 4573  dom cdm 5611  Ord word 6300  Oncon0 6301  ⟶wf 6472  ‘cfv 6476  1_oc1o 8373  2_oc2o 8374   No csur 27573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-ord 6304  df-on 6305  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-1o 8380  df-2o 8381  df-no 27576

This theorem is referenced by:  nosepon  27599