Theorem ordnbtwn 2316

Description: There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41.

Assertion

Ref	Expression
ordnbtwn	|- (Ord A -> -. (A e. B /\ B e. suc A))

Proof of Theorem ordnbtwn

Step	Hyp	Ref	Expression
1		ordn2lp 2219	. . . 4 |- (Ord A -> -. (A e. B /\ B e. A))
2		ordeirr 2217	. . . 4 |- (Ord A -> -. A e. A)
3	1, 2	jca 236	. . 3 |- (Ord A -> (-. (A e. B /\ B e. A) /\ -. A e. A))
4		ioran 254	. . 3 |- (-. ((A e. B /\ B e. A) \/ A e. A) <-> (-. (A e. B /\ B e. A) /\ -. A e. A))
5	3, 4	sylibr 175	. 2 |- (Ord A -> -. ((A e. B /\ B e. A) \/ A e. A))
6		elsuci 2289	. . . . 5 |- (B e. suc A -> (B e. A \/ B = A))
7	6	anim2i 270	. . . 4 |- ((A e. B /\ B e. suc A) -> (A e. B /\ (B e. A \/ B = A)))
8		andi 456	. . . 4 |- ((A e. B /\ (B e. A \/ B = A)) <-> ((A e. B /\ B e. A) \/ (A e. B /\ B = A)))
9	7, 8	sylib 173	. . 3 |- ((A e. B /\ B e. suc A) -> ((A e. B /\ B e. A) \/ (A e. B /\ B = A)))
10		eleq2 1150	. . . . 5 |- (B = A -> (A e. B <-> A e. A))
11	10	biimpac 326	. . . 4 |- ((A e. B /\ B = A) -> A e. A)
12	11	orim2i 273	. . 3 |- (((A e. B /\ B e. A) \/ (A e. B /\ B = A)) -> ((A e. B /\ B e. A) \/ A e. A))
13	9, 12	syl 12	. 2 |- ((A e. B /\ B e. suc A) -> ((A e. B /\ B e. A) \/ A e. A))
14	5, 13	nsyl 102	1 |- (Ord A -> -. (A e. B /\ B e. suc A))

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092  Ord word 2198  suc csuc 2201

This theorem is referenced by:  onnbtwn 2317  ordsucss 2320

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-fr 2169  df-we 2186  df-ord 2202  df-suc 2205

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