Theorem xp01disj 6666

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

Assertion

Ref	Expression
xp01disj	|- ( ( A X. { (/) } ) i^i ( C X. { 1o } ) ) = (/)

Proof of Theorem xp01disj

Step	Hyp	Ref	Expression
1		1n0 6665	. . 3 |- 1o =/= (/)
2	1	necomi 2497	. 2 |- (/) =/= 1o
3		xpsndisj 5189	. 2 |- ( (/) =/= 1o -> ( ( A X. { (/) } ) i^i ( C X. { 1o } ) ) = (/) )
4	2, 3	ax-mp 5	1 |- ( ( A X. { (/) } ) i^i ( C X. { 1o } ) ) = (/)

Syntax hints:   = wceq 1398   =/= wne 2412   i^i cin 3210   (/)c0 3508   {csn 3689   X. cxp 4747   1oc1o 6640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-1o 6647

This theorem is referenced by:  endisj  7075