Theorem unfiexmid 7043

Description: If the union of any two finite sets is finite, excluded middle follows. Remark 8.1.17 of [AczelRathjen], p. 74. (Contributed by Mario Carneiro and Jim Kingdon, 5-Mar-2022.)

Hypothesis

Ref	Expression
unfiexmid.1	|- ( ( x e. Fin /\ y e. Fin ) -> ( x u. y ) e. Fin )

Assertion

Ref	Expression
unfiexmid	|- ( ph \/ -. ph )

Distinct variable group:   ph, x, y

Proof of Theorem unfiexmid

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pr 3651	. . . . 5 |- { { z e. 1o | ph } , 1o } = ( { { z e. 1o | ph } } u. { 1o } )
2		unfiexmid.1	. . . . . . 7 |- ( ( x e. Fin /\ y e. Fin ) -> ( x u. y ) e. Fin )
3	2	rgen2a 2562	. . . . . 6 |- A. x e. Fin A. y e. Fin ( x u. y ) e. Fin
4		df1o2 6540	. . . . . . . . . 10 |- 1o = { (/) }
5		rabeq 2769	. . . . . . . . . 10 |- ( 1o = { (/) } -> { z e. 1o | ph } = { z e. { (/) } | ph } )
6	4, 5	ax-mp 5	. . . . . . . . 9 |- { z e. 1o | ph } = { z e. { (/) } | ph }
7		ordtriexmidlem 4586	. . . . . . . . 9 |- { z e. { (/) } | ph } e. On
8	6, 7	eqeltri 2280	. . . . . . . 8 |- { z e. 1o | ph } e. On
9		snfig 6932	. . . . . . . 8 |- ( { z e. 1o | ph } e. On -> { { z e. 1o | ph } } e. Fin )
10	8, 9	ax-mp 5	. . . . . . 7 |- { { z e. 1o | ph } } e. Fin
11		1onn 6631	. . . . . . . 8 |- 1o e. om
12		snfig 6932	. . . . . . . 8 |- ( 1o e. om -> { 1o } e. Fin )
13	11, 12	ax-mp 5	. . . . . . 7 |- { 1o } e. Fin
14		uneq1 3329	. . . . . . . . 9 |- ( x = { { z e. 1o | ph } } -> ( x u. y ) = ( { { z e. 1o | ph } } u. y ) )
15	14	eleq1d 2276	. . . . . . . 8 |- ( x = { { z e. 1o | ph } } -> ( ( x u. y ) e. Fin <-> ( { { z e. 1o | ph } } u. y ) e. Fin ) )
16		uneq2 3330	. . . . . . . . 9 |- ( y = { 1o } -> ( { { z e. 1o | ph } } u. y ) = ( { { z e. 1o | ph } } u. { 1o } ) )
17	16	eleq1d 2276	. . . . . . . 8 |- ( y = { 1o } -> ( ( { { z e. 1o | ph } } u. y ) e. Fin <-> ( { { z e. 1o | ph } } u. { 1o } ) e. Fin ) )
18	15, 17	rspc2v 2898	. . . . . . 7 |- ( ( { { z e. 1o | ph } } e. Fin /\ { 1o } e. Fin ) -> ( A. x e. Fin A. y e. Fin ( x u. y ) e. Fin -> ( { { z e. 1o | ph } } u. { 1o } ) e. Fin ) )
19	10, 13, 18	mp2an 426	. . . . . 6 |- ( A. x e. Fin A. y e. Fin ( x u. y ) e. Fin -> ( { { z e. 1o | ph } } u. { 1o } ) e. Fin )
20	3, 19	ax-mp 5	. . . . 5 |- ( { { z e. 1o | ph } } u. { 1o } ) e. Fin
21	1, 20	eqeltri 2280	. . . 4 |- { { z e. 1o | ph } , 1o } e. Fin
22	8	elexi 2790	. . . . 5 |- { z e. 1o | ph } e. _V
23	22	prid1 3750	. . . 4 |- { z e. 1o | ph } e. { { z e. 1o | ph } , 1o }
24	11	elexi 2790	. . . . 5 |- 1o e. _V
25	24	prid2 3751	. . . 4 |- 1o e. { { z e. 1o | ph } , 1o }
26		fidceq 6994	. . . 4 |- ( ( { { z e. 1o | ph } , 1o } e. Fin /\ { z e. 1o | ph } e. { { z e. 1o | ph } , 1o } /\ 1o e. { { z e. 1o | ph } , 1o } ) -> DECID { z e. 1o | ph } = 1o )
27	21, 23, 25, 26	mp3an 1350	. . 3 |- DECID { z e. 1o | ph } = 1o
28		exmiddc 838	. . 3 |- (DECID { z e. 1o | ph } = 1o -> ( { z e. 1o | ph } = 1o \/ -. { z e. 1o | ph } = 1o ) )
29	27, 28	ax-mp 5	. 2 |- ( { z e. 1o | ph } = 1o \/ -. { z e. 1o | ph } = 1o )
30	4	eqeq2i 2218	. . . 4 |- ( { z e. 1o | ph } = 1o <-> { z e. 1o | ph } = { (/) } )
31		0ex 4188	. . . . 5 |- (/) e. _V
32		biidd 172	. . . . 5 |- ( z = (/) -> ( ph <-> ph ) )
33	31, 32	rabsnt 3719	. . . 4 |- ( { z e. 1o | ph } = { (/) } -> ph )
34	30, 33	sylbi 121	. . 3 |- ( { z e. 1o | ph } = 1o -> ph )
35		df-rab 2495	. . . . 5 |- { z e. 1o | ph } = { z | ( z e. 1o /\ ph ) }
36		iba 300	. . . . . 6 |- ( ph -> ( z e. 1o <-> ( z e. 1o /\ ph ) ) )
37	36	abbi2dv 2326	. . . . 5 |- ( ph -> 1o = { z | ( z e. 1o /\ ph ) } )
38	35, 37	eqtr4id 2259	. . . 4 |- ( ph -> { z e. 1o | ph } = 1o )
39	38	con3i 633	. . 3 |- ( -. { z e. 1o | ph } = 1o -> -. ph )
40	34, 39	orim12i 761	. 2 |- ( ( { z e. 1o | ph } = 1o \/ -. { z e. 1o | ph } = 1o ) -> ( ph \/ -. ph ) )
41	29, 40	ax-mp 5	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710  DECID wdc 836   = wceq 1373   e. wcel 2178   {cab 2193   A.wral 2486   {crab 2490   u. cun 3173   (/)c0 3469   {csn 3644   {cpr 3645   Oncon0 4429   omcom 4657   1oc1o 6520   Fincfn 6852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-nul 4187  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-iinf 4655

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2779  df-sbc 3007  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-nul 3470  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-br 4061  df-opab 4123  df-tr 4160  df-id 4359  df-iord 4432  df-on 4434  df-suc 4437  df-iom 4658  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-f1o 5298  df-fv 5299  df-1o 6527  df-en 6853  df-fin 6855

This theorem is referenced by: (None)