Theorem oa00 4186

Description: An ordinal sum is zero iff both of its arguments are zero.

Assertion

Ref	Expression
oa00	|- ((A e. On /\ B e. On) -> ((A +o B) = (/) <-> (A = (/) /\ B = (/))))

Proof of Theorem oa00

Step	Hyp	Ref	Expression
1		on0eln0 3020	. . . . . . 7 |- (A e. On -> ((/) e. A <-> A =/= (/)))
2	1	adantr 389	. . . . . 6 |- ((A e. On /\ B e. On) -> ((/) e. A <-> A =/= (/)))
3		oaword1 4179	. . . . . . 7 |- ((A e. On /\ B e. On) -> A (_ (A +o B))
4	3	sseld 2064	. . . . . 6 |- ((A e. On /\ B e. On) -> ((/) e. A -> (/) e. (A +o B)))
5	2, 4	sylbird 205	. . . . 5 |- ((A e. On /\ B e. On) -> (A =/= (/) -> (/) e. (A +o B)))
6		ne0i 2283	. . . . 5 |- ((/) e. (A +o B) -> (A +o B) =/= (/))
7	5, 6	syl6 22	. . . 4 |- ((A e. On /\ B e. On) -> (A =/= (/) -> (A +o B) =/= (/)))
8	7	necon4d 1626	. . 3 |- ((A e. On /\ B e. On) -> ((A +o B) = (/) -> A = (/)))
9		on0eln0 3020	. . . . . . 7 |- (B e. On -> ((/) e. B <-> B =/= (/)))
10	9	adantl 388	. . . . . 6 |- ((A e. On /\ B e. On) -> ((/) e. B <-> B =/= (/)))
11		0elon 3018	. . . . . . . 8 |- (/) e. On
12		oaord 4174	. . . . . . . 8 |- (((/) e. On /\ B e. On /\ A e. On) -> ((/) e. B <-> (A +o (/)) e. (A +o B)))
13	11, 12	mp3an1 902	. . . . . . 7 |- ((B e. On /\ A e. On) -> ((/) e. B <-> (A +o (/)) e. (A +o B)))
14	13	ancoms 436	. . . . . 6 |- ((A e. On /\ B e. On) -> ((/) e. B <-> (A +o (/)) e. (A +o B)))
15	10, 14	bitr3d 529	. . . . 5 |- ((A e. On /\ B e. On) -> (B =/= (/) <-> (A +o (/)) e. (A +o B)))
16		ne0i 2283	. . . . 5 |- ((A +o (/)) e. (A +o B) -> (A +o B) =/= (/))
17	15, 16	syl6bi 214	. . . 4 |- ((A e. On /\ B e. On) -> (B =/= (/) -> (A +o B) =/= (/)))
18	17	necon4d 1626	. . 3 |- ((A e. On /\ B e. On) -> ((A +o B) = (/) -> B = (/)))
19	8, 18	jcad 599	. 2 |- ((A e. On /\ B e. On) -> ((A +o B) = (/) -> (A = (/) /\ B = (/))))
20		opreq12 3965	. . 3 |- ((A = (/) /\ B = (/)) -> (A +o B) = ((/) +o (/)))
21		oa0 4148	. . . 4 |- ((/) e. On -> ((/) +o (/)) = (/))
22	11, 21	ax-mp 7	. . 3 |- ((/) +o (/)) = (/)
23	20, 22	syl6eq 1521	. 2 |- ((A = (/) /\ B = (/)) -> (A +o B) = (/))
24	19, 23	impbid1 516	1 |- ((A e. On /\ B e. On) -> ((A +o B) = (/) <-> (A = (/) /\ B = (/))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957   =/= wne 1583  (/)c0 2277  Oncon0 2944  (class class class)co 3958   +o coa 4123

This theorem is referenced by:  oalimcl 4187

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-oadd 4128

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