Theorem xp01disjl 6489

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 6487	. . 3 |- 1o =/= (/)
2	1	necomi 2449	. 2 |- (/) =/= 1o
3		disjsn2 3682	. 2 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
4		xpdisj1 5091	. 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 1364   =/= wne 2364   i^i cin 3153   (/)c0 3447   {csn 3619   X. cxp 4658   1oc1o 6464

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-opab 4092  df-suc 4403  df-xp 4666  df-rel 4667  df-1o 6471

This theorem is referenced by:  djucomen  7278  djuassen  7279  xpdjuen  7280