Theorem xp01disjl 6707

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 6705	. . 3 |- 1o =/= (/)
2	1	necomi 2505	. 2 |- (/) =/= 1o
3		disjsn2 3772	. 2 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
4		xpdisj1 5212	. 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 ) ) = (/)

This proof depends on syntax axioms:   = wceq 1402   =/= wne 2420   i^i cin 3219   (/)c0 3520   {csn 3709   X. cxp 4772   1oc1o 6680

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-opab 4193  df-suc 4516  df-xp 4780  df-rel 4781  df-1o 6687

This theorem is used by:  djucomen  7572  djuassen  7573  xpdjuen  7574