Theorem weinxp 5732

Description: Intersection of well-ordering with Cartesian product of its field. (Contributed by NM, 9-Mar-1997.) (Revised by Mario Carneiro, 10-Jul-2014.)

Assertion

Ref	Expression
weinxp	⊢ (𝑅 We 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) We 𝐴)

Proof of Theorem weinxp

Step	Hyp	Ref	Expression
1		frinxp 5730	. . 3 ⊢ (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)
2		soinxp 5729	. . 3 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)
3	1, 2	anbi12i 640	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴 ∧ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴))
4		df-we 5602	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
5		df-we 5602	. 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) We 𝐴 ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴 ∧ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴))
6	3, 4, 5	3bitr4i 306	1 ⊢ (𝑅 We 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) We 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∩ cin 3897   Or wor 5554   Fr wfr 5597   We wwe 5599   × cxp 5645

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653

This theorem is used by:  wemapwe  9676  infxpenlem  10064  dfac8b  10082  ac10ct  10085  canthwelem  10707  ltbwe  22315  vitali  25896  fin2so  38450  dnwech  43993  aomclem5  44003