Theorem ordsseleq 2976

Description: For ordinal classes, subclass is equivalent to membership or equality.

Assertion

Ref	Expression
ordsseleq	|- ((Ord A /\ Ord B) -> (A (_ B <-> (A e. B \/ A = B)))

Proof of Theorem ordsseleq

Step	Hyp	Ref	Expression
1		ordelssne 2974	. . . . . . . 8 |- ((Ord A /\ Ord B) -> (A e. B <-> (A (_ B /\ A =/= B)))
2	1	biimprd 154	. . . . . . 7 |- ((Ord A /\ Ord B) -> ((A (_ B /\ A =/= B) -> A e. B))
3	2	exp3a 375	. . . . . 6 |- ((Ord A /\ Ord B) -> (A (_ B -> (A =/= B -> A e. B)))
4	3	imp 350	. . . . 5 |- (((Ord A /\ Ord B) /\ A (_ B) -> (A =/= B -> A e. B))
5	4	necon1bd 1632	. . . 4 |- (((Ord A /\ Ord B) /\ A (_ B) -> (-. A e. B -> A = B))
6	5	orrd 233	. . 3 |- (((Ord A /\ Ord B) /\ A (_ B) -> (A e. B \/ A = B))
7	6	ex 373	. 2 |- ((Ord A /\ Ord B) -> (A (_ B -> (A e. B \/ A = B)))
8		pm3.26 319	. . . . 5 |- ((A (_ B /\ A =/= B) -> A (_ B)
9	1, 8	syl6bi 214	. . . 4 |- ((Ord A /\ Ord B) -> (A e. B -> A (_ B))
10		eqimss 2109	. . . 4 |- (A = B -> A (_ B)
11	9, 10	jctir 293	. . 3 |- ((Ord A /\ Ord B) -> ((A e. B -> A (_ B) /\ (A = B -> A (_ B)))
12		jaob 422	. . 3 |- (((A e. B \/ A = B) -> A (_ B) <-> ((A e. B -> A (_ B) /\ (A = B -> A (_ B)))
13	11, 12	sylibr 200	. 2 |- ((Ord A /\ Ord B) -> ((A e. B \/ A = B) -> A (_ B))
14	7, 13	impbid 516	1 |- ((Ord A /\ Ord B) -> (A (_ B <-> (A e. B \/ A = B)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585   (_ wss 2047  Ord word 2947

This theorem is referenced by:  ordtri3or 2979  ordtri1 2980  ordtri2 2982  onsseleq 2999  ordtr2 3002  ordsssuc 3057  ordsucelsuc 3073  ordtri2or 3077  limom 3146

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951

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