Theorem oaordex 5240

Description: Existence theorem for ordering of ordinal sum. Similar to Proposition 4.34(f) of [Mendelson] p. 266 and its converse.

Assertion

Ref	Expression
oaordex	|- ((A e. On /\ B e. On) -> (A e. B <-> E.x e. On ((/) e. x /\ (A +o x) = B)))

Distinct variable groups:   x,A   x,B

Proof of Theorem oaordex

Step	Hyp	Ref	Expression
1		onelss 3705	. . . . 5 |- (B e. On -> (A e. B -> A C_ B))
2	1	adantl 424	. . . 4 |- ((A e. On /\ B e. On) -> (A e. B -> A C_ B))
3		oawordex 5239	. . . 4 |- ((A e. On /\ B e. On) -> (A C_ B <-> E.x e. On (A +o x) = B))
4	2, 3	sylibd 219	. . 3 |- ((A e. On /\ B e. On) -> (A e. B -> E.x e. On (A +o x) = B))
5		oaord1 5233	. . . . . . . . . . . . 13 |- ((A e. On /\ x e. On) -> ((/) e. x <-> A e. (A +o x)))
6		eleq2 1958	. . . . . . . . . . . . 13 |- ((A +o x) = B -> (A e. (A +o x) <-> A e. B))
7	5, 6	sylan9bb 599	. . . . . . . . . . . 12 |- (((A e. On /\ x e. On) /\ (A +o x) = B) -> ((/) e. x <-> A e. B))
8	7	biimprcd 173	. . . . . . . . . . 11 |- (A e. B -> (((A e. On /\ x e. On) /\ (A +o x) = B) -> (/) e. x))
9	8	exp4c 411	. . . . . . . . . 10 |- (A e. B -> (A e. On -> (x e. On -> ((A +o x) = B -> (/) e. x))))
10	9	com12 14	. . . . . . . . 9 |- (A e. On -> (A e. B -> (x e. On -> ((A +o x) = B -> (/) e. x))))
11	10	imp4b 392	. . . . . . . 8 |- ((A e. On /\ A e. B) -> ((x e. On /\ (A +o x) = B) -> (/) e. x))
12		simpr 350	. . . . . . . . 9 |- ((x e. On /\ (A +o x) = B) -> (A +o x) = B)
13	12	a1i 8	. . . . . . . 8 |- ((A e. On /\ A e. B) -> ((x e. On /\ (A +o x) = B) -> (A +o x) = B))
14	11, 13	jcad 661	. . . . . . 7 |- ((A e. On /\ A e. B) -> ((x e. On /\ (A +o x) = B) -> ((/) e. x /\ (A +o x) = B)))
15	14	exp3a 405	. . . . . 6 |- ((A e. On /\ A e. B) -> (x e. On -> ((A +o x) = B -> ((/) e. x /\ (A +o x) = B))))
16	15	reximdvai 2201	. . . . 5 |- ((A e. On /\ A e. B) -> (E.x e. On (A +o x) = B -> E.x e. On ((/) e. x /\ (A +o x) = B)))
17	16	ex 402	. . . 4 |- (A e. On -> (A e. B -> (E.x e. On (A +o x) = B -> E.x e. On ((/) e. x /\ (A +o x) = B))))
18	17	adantr 425	. . 3 |- ((A e. On /\ B e. On) -> (A e. B -> (E.x e. On (A +o x) = B -> E.x e. On ((/) e. x /\ (A +o x) = B))))
19	4, 18	mpdd 57	. 2 |- ((A e. On /\ B e. On) -> (A e. B -> E.x e. On ((/) e. x /\ (A +o x) = B)))
20	7	biimpd 170	. . . . . . 7 |- (((A e. On /\ x e. On) /\ (A +o x) = B) -> ((/) e. x -> A e. B))
21	20	exp31 407	. . . . . 6 |- (A e. On -> (x e. On -> ((A +o x) = B -> ((/) e. x -> A e. B))))
22	21	com34 40	. . . . 5 |- (A e. On -> (x e. On -> ((/) e. x -> ((A +o x) = B -> A e. B))))
23	22	imp4a 391	. . . 4 |- (A e. On -> (x e. On -> (((/) e. x /\ (A +o x) = B) -> A e. B)))
24	23	r19.23adv 2215	. . 3 |- (A e. On -> (E.x e. On ((/) e. x /\ (A +o x) = B) -> A e. B))
25	24	adantr 425	. 2 |- ((A e. On /\ B e. On) -> (E.x e. On ((/) e. x /\ (A +o x) = B) -> A e. B))
26	19, 25	impbid 574	1 |- ((A e. On /\ B e. On) -> (A e. B <-> E.x e. On ((/) e. x /\ (A +o x) = B)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106   C_ wss 2593  (/)c0 2875  Oncon0 3657  (class class class)co 4884   +o coa 5174

This theorem is referenced by:  nnaordex 5306

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-13 1311  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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  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-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-oadd 5179

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