Theorem oaordex 4176

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		onelsst 2990	. . . . 5 |- (B e. On -> (A e. B -> A (_ B))
2	1	adantl 388	. . . 4 |- ((A e. On /\ B e. On) -> (A e. B -> A (_ B))
3		oawordex 4175	. . . 4 |- ((A e. On /\ B e. On) -> (A (_ B <-> E.x e. On (A +o x) = B))
4	2, 3	sylibd 202	. . 3 |- ((A e. On /\ B e. On) -> (A e. B -> E.x e. On (A +o x) = B))
5		oaord1 4169	. . . . . . . . . . . . 13 |- ((A e. On /\ x e. On) -> ((/) e. x <-> A e. (A +o x)))
6		eleq2 1527	. . . . . . . . . . . . 13 |- ((A +o x) = B -> (A e. (A +o x) <-> A e. B))
7	5, 6	sylan9bb 538	. . . . . . . . . . . 12 |- (((A e. On /\ x e. On) /\ (A +o x) = B) -> ((/) e. x <-> A e. B))
8	7	biimprcd 156	. . . . . . . . . . 11 |- (A e. B -> (((A e. On /\ x e. On) /\ (A +o x) = B) -> (/) e. x))
9	8	exp4c 380	. . . . . . . . . 10 |- (A e. B -> (A e. On -> (x e. On -> ((A +o x) = B -> (/) e. x))))
10	9	com12 11	. . . . . . . . 9 |- (A e. On -> (A e. B -> (x e. On -> ((A +o x) = B -> (/) e. x))))
11	10	imp4b 365	. . . . . . . 8 |- ((A e. On /\ A e. B) -> ((x e. On /\ (A +o x) = B) -> (/) e. x))
12		pm3.27 323	. . . . . . . . 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 598	. . . . . . 7 |- ((A e. On /\ A e. B) -> ((x e. On /\ (A +o x) = B) -> ((/) e. x /\ (A +o x) = B)))
15	14	exp3a 375	. . . . . 6 |- ((A e. On /\ A e. B) -> (x e. On -> ((A +o x) = B -> ((/) e. x /\ (A +o x) = B))))
16	15	r19.22dv 1729	. . . . 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 373	. . . 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 389	. . 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 46	. 2 |- ((A e. On /\ B e. On) -> (A e. B -> E.x e. On ((/) e. x /\ (A +o x) = B)))
20	7	biimpd 153	. . . . . . 7 |- (((A e. On /\ x e. On) /\ (A +o x) = B) -> ((/) e. x -> A e. B))
21	20	exp31 376	. . . . . 6 |- (A e. On -> (x e. On -> ((A +o x) = B -> ((/) e. x -> A e. B))))
22	21	com34 36	. . . . 5 |- (A e. On -> (x e. On -> ((/) e. x -> ((A +o x) = B -> A e. B))))
23	22	imp4a 364	. . . 4 |- (A e. On -> (x e. On -> (((/) e. x /\ (A +o x) = B) -> A e. B)))
24	23	r19.23adv 1738	. . 3 |- (A e. On -> (E.x e. On ((/) e. x /\ (A +o x) = B) -> A e. B))
25	24	adantr 389	. 2 |- ((A e. On /\ B e. On) -> (E.x e. On ((/) e. x /\ (A +o x) = B) -> A e. B))
26	19, 25	impbid 514	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 146   /\ wa 223   = wceq 953   e. wcel 955  E.wrex 1638   (_ wss 2037  (/)c0 2270  Oncon0 2938  (class class class)co 3948   +o coa 4114

This theorem is referenced by:  nnaordex 4233

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-oadd 4119

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