Theorem ltpiord 10778

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 10766	. . 3 ⊢ <N = ( E ∩ (N × N))
2	1	breqi 5095	. 2 ⊢ (𝐴 <N 𝐵 ↔ 𝐴( E ∩ (N × N))𝐵)
3		brinxp 5693	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 E 𝐵 ↔ 𝐴( E ∩ (N × N))𝐵))
4		epelg 5515	. . . 4 ⊢ (𝐵 ∈ N → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
5	4	adantl 481	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
6	3, 5	bitr3d 281	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴( E ∩ (N × N))𝐵 ↔ 𝐴 ∈ 𝐵))
7	2, 6	bitrid 283	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2111   ∩ cin 3896   class class class wbr 5089   E cep 5513   × cxp 5612  Ncnpi 10735   <_N clti 10738

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-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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-sb 2068  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 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-eprel 5514  df-xp 5620  df-lti 10766

This theorem is referenced by:  ltexpi  10793  ltapi  10794  ltmpi  10795  1lt2pi  10796  nlt1pi  10797  indpi  10798  nqereu  10820