Theorem ltpiord 10805

Description: Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
ltpiord	⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵))

Proof of Theorem ltpiord

Step	Hyp	Ref	Expression
1		df-lti 10793	. . 3 ⊢ <N = ( E ∩ (N × N))
2	1	breqi 5081	. 2 ⊢ (𝐴 <N 𝐵 ↔ 𝐴( E ∩ (N × N))𝐵)
3		brinxp 5700	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 E 𝐵 ↔ 𝐴( E ∩ (N × N))𝐵))
4		epelg 5522	. . . 4 ⊢ (𝐵 ∈ N → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
5	4	adantl 483	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
6	3, 5	bitr3d 283	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴( E ∩ (N × N))𝐵 ↔ 𝐴 ∈ 𝐵))
7	2, 6	bitrid 285	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∈ wcel 2121   ∩ cin 3884   class class class wbr 5075   E cep 5520   × cxp 5619  Ncnpi 10762   <_N clti 10765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-eprel 5521  df-xp 5627  df-lti 10793

This theorem is referenced by:  ltexpi  10820  ltapi  10821  ltmpi  10822  1lt2pi  10823  nlt1pi  10824  indpi  10825  nqereu  10847