Theorem xp01disj 7576

Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)

Assertion

Ref	Expression
xp01disj	⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅

Proof of Theorem xp01disj

Step	Hyp	Ref	Expression
1		1n0 7575	. . 3 ⊢ 1𝑜 ≠ ∅
2	1	necomi 2848	. 2 ⊢ ∅ ≠ 1𝑜
3		xpsndisj 5557	. 2 ⊢ (∅ ≠ 1𝑜 → ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅

Syntax hints:   = wceq 1483   ≠ wne 2794   ∩ cin 3573  ∅c0 3915  {csn 4177   × cxp 5112  1_𝑜c1o 7553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-suc 5729  df-1o 7560

This theorem is referenced by:  endisj  8047  uncdadom  8993  cdaun  8994  cdaen  8995  cda1dif  8998  pm110.643  8999  cdacomen  9003  cdaassen  9004  xpcdaen  9005  mapcdaen  9006  cdadom1  9008  infcda1  9015