Theorem xp01disjl 6645

Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)

Assertion

Ref	Expression
xp01disjl	|- ( ( { (/) } X. A ) i^i ( { 1o } X. C ) ) = (/)

Proof of Theorem xp01disjl

Step	Hyp	Ref	Expression
1		1n0 6643	. . 3 |- 1o =/= (/)
2	1	necomi 2488	. 2 |- (/) =/= 1o
3		disjsn2 3736	. 2 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
4		xpdisj1 5168	. 2 |- ( ( { (/) } i^i { 1o } ) = (/) -> ( ( { (/) } X. A ) i^i ( { 1o } X. C ) ) = (/) )
5	2, 3, 4	mp2b 8	1 |- ( ( { (/) } X. A ) i^i ( { 1o } X. C ) ) = (/)

Syntax hints:   = wceq 1398   =/= wne 2403   i^i cin 3200   (/)c0 3496   {csn 3673   X. cxp 4729   1oc1o 6618

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 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-suc 4474  df-xp 4737  df-rel 4738  df-1o 6625

This theorem is referenced by:  djucomen  7491  djuassen  7492  xpdjuen  7493