Theorem xp01disjl 6680

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 6678	. . 3 |- 1o =/= (/)
2	1	necomi 2499	. 2 |- (/) =/= 1o
3		disjsn2 3757	. 2 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
4		xpdisj1 5192	. 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 2414   i^i cin 3213   (/)c0 3512   {csn 3694   X. cxp 4752   1oc1o 6653

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 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-opab 4177  df-suc 4497  df-xp 4760  df-rel 4761  df-1o 6660

This theorem is referenced by:  djucomen  7536  djuassen  7537  xpdjuen  7538