Theorem nlt1pi 3827

Description: No positive integer is less than one.

Assertion

Ref	Expression
nlt1pi	⊢ ¬ A <N 1o

Proof of Theorem nlt1pi

Step	Hyp	Ref	Expression
1		elni 3798	. . . 4 ⊢ (A ∈ N ↔ (A ∈ ω ∧ ¬ A = ∅))
2	1	pm3.27bd 263	. . 3 ⊢ (A ∈ N → ¬ A = ∅)
3		noel 1711	. . . . 5 ⊢ ¬ A ∈ ∅
4		1pi 3805	. . . . . . . . 9 ⊢ 1o ∈ N
5		ltpiord 3809	. . . . . . . . 9 ⊢ ((A ∈ N ∧ 1o ∈ N) → (A <N 1o ↔ A ∈ 1o))
6	4, 5	mpan2 519	. . . . . . . 8 ⊢ (A ∈ N → (A <N 1o ↔ A ∈ 1o))
7		elsucg 2290	. . . . . . . . 9 ⊢ (A ∈ N → (A ∈ suc ∅ ↔ (A ∈ ∅ ∨ A = ∅)))
8		df-1o 3104	. . . . . . . . . 10 ⊢ 1o = suc ∅
9	8	eleq2i 1153	. . . . . . . . 9 ⊢ (A ∈ 1o ↔ A ∈ suc ∅)
10	7, 9	syl5bb 410	. . . . . . . 8 ⊢ (A ∈ N → (A ∈ 1o ↔ (A ∈ ∅ ∨ A = ∅)))
11	6, 10	bitrd 406	. . . . . . 7 ⊢ (A ∈ N → (A <N 1o ↔ (A ∈ ∅ ∨ A = ∅)))
12	11	biimpa 324	. . . . . 6 ⊢ ((A ∈ N ∧ A <N 1o) → (A ∈ ∅ ∨ A = ∅))
13	12	ord 202	. . . . 5 ⊢ ((A ∈ N ∧ A <N 1o) → (¬ A ∈ ∅ → A = ∅))
14	3, 13	mpi 44	. . . 4 ⊢ ((A ∈ N ∧ A <N 1o) → A = ∅)
15	14	exp 291	. . 3 ⊢ (A ∈ N → (A <N 1o → A = ∅))
16	2, 15	mtod 95	. 2 ⊢ (A ∈ N → ¬ A <N 1o)
17	4	elisseti 1355	. . . . 5 ⊢ 1o ∈ V
18		ltrelpi 3811	. . . . 5 ⊢ <N ⊆ (N × N)
19	17, 18	brel 2459	. . . 4 ⊢ (A <N 1o → (A ∈ N ∧ 1o ∈ N))
20	19	pm3.26d 258	. . 3 ⊢ (A <N 1o → A ∈ N)
21	20	con3i 90	. 2 ⊢ (¬ A ∈ N → ¬ A <N 1o)
22	16, 21	pm2.61i 110	1 ⊢ ¬ A <N 1o

Syntax hints:  ¬ wn 1   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∅c0 1707   class class class wbr 2054  suc csuc 2201  ωcom 2372  1_oc1o 3099  Ncnpi 3766   <_N clti 3769

This theorem is referenced by:  indpi 3828  prlem934b 3932

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-1o 3104  df-ni 3794  df-lti 3797

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