Theorem sspred 13886

Description: Another subset/predecessor class relationship.

Assertion

Ref	Expression
sspred	|- ((B C_ A /\ Pred(R, A, X) C_ B) -> Pred(R, A, X) = Pred(R, B, X))

Proof of Theorem sspred

Step	Hyp	Ref	Expression
1		ineq1 2789	. . . . 5 |- ((A i^i B) = B -> ((A i^i B) i^i (`'R"{X})) = (B i^i (`'R"{X})))
2	1	eqeq1d 1892	. . . 4 |- ((A i^i B) = B -> (((A i^i B) i^i (`'R"{X})) = (A i^i (`'R"{X})) <-> (B i^i (`'R"{X})) = (A i^i (`'R"{X}))))
3	2	biimpa 460	. . 3 |- (((A i^i B) = B /\ ((A i^i B) i^i (`'R"{X})) = (A i^i (`'R"{X}))) -> (B i^i (`'R"{X})) = (A i^i (`'R"{X})))
4		df-pred 13880	. . . . . 6 |- Pred(R, A, X) = (A i^i (`'R"{X}))
5		df-pred 13880	. . . . . 6 |- Pred(R, B, X) = (B i^i (`'R"{X}))
6	4, 5	eqeq12i 1897	. . . . 5 |- (Pred(R, A, X) = Pred(R, B, X) <-> (A i^i (`'R"{X})) = (B i^i (`'R"{X})))
7	6	biimpri 169	. . . 4 |- ((A i^i (`'R"{X})) = (B i^i (`'R"{X})) -> Pred(R, A, X) = Pred(R, B, X))
8	7	eqcoms 1887	. . 3 |- ((B i^i (`'R"{X})) = (A i^i (`'R"{X})) -> Pred(R, A, X) = Pred(R, B, X))
9	3, 8	syl 12	. 2 |- (((A i^i B) = B /\ ((A i^i B) i^i (`'R"{X})) = (A i^i (`'R"{X}))) -> Pred(R, A, X) = Pred(R, B, X))
10		sseqin2 2811	. 2 |- (B C_ A <-> (A i^i B) = B)
11	4	sseq1i 2641	. . 3 |- (Pred(R, A, X) C_ B <-> (A i^i (`'R"{X})) C_ B)
12		df-ss 2605	. . 3 |- ((A i^i (`'R"{X})) C_ B <-> ((A i^i (`'R"{X})) i^i B) = (A i^i (`'R"{X})))
13		in23 2806	. . . 4 |- ((A i^i (`'R"{X})) i^i B) = ((A i^i B) i^i (`'R"{X}))
14	13	eqeq1i 1891	. . 3 |- (((A i^i (`'R"{X})) i^i B) = (A i^i (`'R"{X})) <-> ((A i^i B) i^i (`'R"{X})) = (A i^i (`'R"{X})))
15	11, 12, 14	3bitri 194	. 2 |- (Pred(R, A, X) C_ B <-> ((A i^i B) i^i (`'R"{X})) = (A i^i (`'R"{X})))
16	9, 10, 15	syl2anb 504	1 |- ((B C_ A /\ Pred(R, A, X) C_ B) -> Pred(R, A, X) = Pred(R, B, X))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   i^i cin 2592   C_ wss 2593  {csn 3044  `'ccnv 3985  "cima 3989  Predcpred 13879

This theorem is referenced by:  frmin 13938

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-ss 2605  df-pred 13880

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