Theorem ltpiord 10816

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 10804	. . 3 ⊢ <N = ( E ∩ (N × N))
2	1	breqi 5108	. 2 ⊢ (𝐴 <N 𝐵 ↔ 𝐴( E ∩ (N × N))𝐵)
3		brinxp 5710	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 E 𝐵 ↔ 𝐴( E ∩ (N × N))𝐵))
4		epelg 5532	. . . 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 2109   ∩ cin 3910   class class class wbr 5102   E cep 5530   × cxp 5629  Ncnpi 10773   <_N clti 10776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-eprel 5531  df-xp 5637  df-lti 10804

This theorem is referenced by:  ltexpi  10831  ltapi  10832  ltmpi  10833  1lt2pi  10834  nlt1pi  10835  indpi  10836  nqereu  10858