Theorem sqpeq 14383

Description: A square cross product is an equivalence relation (in general it's not a poset).

Assertion

Ref	Expression
sqpeq	|- Er (A X. A)

Proof of Theorem sqpeq

Step	Hyp	Ref	Expression
1		cnvxp 4332	. . 3 |- `'(A X. A) = (A X. A)
2		cpref 14379	. . 3 |- ((A X. A) o. (A X. A)) C_ (A X. A)
3		uneq1 2748	. . . . 5 |- (`'(A X. A) = (A X. A) -> (`'(A X. A) u. (A X. A)) = ((A X. A) u. (A X. A)))
4		unidm 2743	. . . . . 6 |- ((A X. A) u. (A X. A)) = (A X. A)
5		eqtr 1904	. . . . . . 7 |- (((`'(A X. A) u. (A X. A)) = ((A X. A) u. (A X. A)) /\ ((A X. A) u. (A X. A)) = (A X. A)) -> (`'(A X. A) u. (A X. A)) = (A X. A))
6		sseq2 2639	. . . . . . . 8 |- ((`'(A X. A) u. (A X. A)) = (A X. A) -> ((`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (`'(A X. A) u. (A X. A)) <-> (`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (A X. A)))
7	6	biimpd 170	. . . . . . 7 |- ((`'(A X. A) u. (A X. A)) = (A X. A) -> ((`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (`'(A X. A) u. (A X. A)) -> (`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (A X. A)))
8	5, 7	syl 12	. . . . . 6 |- (((`'(A X. A) u. (A X. A)) = ((A X. A) u. (A X. A)) /\ ((A X. A) u. (A X. A)) = (A X. A)) -> ((`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (`'(A X. A) u. (A X. A)) -> (`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (A X. A)))
9	4, 8	mpan2 760	. . . . 5 |- ((`'(A X. A) u. (A X. A)) = ((A X. A) u. (A X. A)) -> ((`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (`'(A X. A) u. (A X. A)) -> (`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (A X. A)))
10	3, 9	syl 12	. . . 4 |- (`'(A X. A) = (A X. A) -> ((`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (`'(A X. A) u. (A X. A)) -> (`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (A X. A)))
11		unss2 2777	. . . 4 |- (((A X. A) o. (A X. A)) C_ (A X. A) -> (`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (`'(A X. A) u. (A X. A)))
12	10, 11	syl5 20	. . 3 |- (`'(A X. A) = (A X. A) -> (((A X. A) o. (A X. A)) C_ (A X. A) -> (`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (A X. A)))
13	1, 2, 12	mp2 54	. 2 |- (`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (A X. A)
14		df-er 5318	. 2 |- (Er (A X. A) <-> (`'(A X. A) u. ((A X. A) o. (A X. A))) C_ (A X. A))
15	13, 14	mpbir 207	1 |- Er (A X. A)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   u. cun 2591   C_ wss 2593   X. cxp 3984  `'ccnv 3985   o. ccom 3990  Er wer 5315

This theorem is referenced by:  tostos 14637

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-er 5318

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