Theorem xp01disjl 6331

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 6329	. . 3 |- 1o =/= (/)
2	1	necomi 2393	. 2 |- (/) =/= 1o
3		disjsn2 3586	. 2 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
4		xpdisj1 4963	. 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 1331   =/= wne 2308   i^i cin 3070   (/)c0 3363   {csn 3527   X. cxp 4537   1oc1o 6306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-suc 4293  df-xp 4545  df-rel 4546  df-1o 6313

This theorem is referenced by:  djucomen  7072  djuassen  7073  xpdjuen  7074