Theorem ordnbtwn 3058

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 2963	. . . 4 |- (Ord A -> -. (A e. B /\ B e. A))
2		ordirr 2961	. . . 4 |- (Ord A -> -. A e. A)
3	1, 2	jca 288	. . 3 |- (Ord A -> (-. (A e. B /\ B e. A) /\ -. A e. A))
4		ioran 306	. . 3 |- (-. ((A e. B /\ B e. A) \/ A e. A) <-> (-. (A e. B /\ B e. A) /\ -. A e. A))
5	3, 4	sylibr 200	. 2 |- (Ord A -> -. ((A e. B /\ B e. A) \/ A e. A))
6		elsuci 3030	. . . . 5 |- (B e. suc A -> (B e. A \/ B = A))
7	6	anim2i 335	. . . 4 |- ((A e. B /\ B e. suc A) -> (A e. B /\ (B e. A \/ B = A)))
8		andi 603	. . . 4 |- ((A e. B /\ (B e. A \/ B = A)) <-> ((A e. B /\ B e. A) \/ (A e. B /\ B = A)))
9	7, 8	sylib 198	. . 3 |- ((A e. B /\ B e. suc A) -> ((A e. B /\ B e. A) \/ (A e. B /\ B = A)))
10		eleq2 1532	. . . . 5 |- (B = A -> (A e. B <-> A e. A))
11	10	biimpac 418	. . . 4 |- ((A e. B /\ B = A) -> A e. A)
12	11	orim2i 338	. . 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 10	. 2 |- ((A e. B /\ B e. suc A) -> ((A e. B /\ B e. A) \/ A e. A))
14	5, 13	nsyl 116	1 |- (Ord A -> -. (A e. B /\ B e. suc A))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   = wceq 954   e. wcel 956  Ord word 2942  suc csuc 2945

This theorem is referenced by:  onnbtwn 3059  ordsucss 3064

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-fr 2912  df-we 2929  df-ord 2946  df-suc 2949

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