Theorem unfiexmid 7080

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 3673	. . . . 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 2584	. . . . . 6 |- A. x e. Fin A. y e. Fin ( x u. y ) e. Fin
4		df1o2 6575	. . . . . . . . . 10 |- 1o = { (/) }
5		rabeq 2791	. . . . . . . . . 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 4611	. . . . . . . . 9 |- { z e. { (/) } | ph } e. On
8	6, 7	eqeltri 2302	. . . . . . . 8 |- { z e. 1o | ph } e. On
9		snfig 6967	. . . . . . . 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 6666	. . . . . . . 8 |- 1o e. om
12		snfig 6967	. . . . . . . 8 |- ( 1o e. om -> { 1o } e. Fin )
13	11, 12	ax-mp 5	. . . . . . 7 |- { 1o } e. Fin
14		uneq1 3351	. . . . . . . . 9 |- ( x = { { z e. 1o | ph } } -> ( x u. y ) = ( { { z e. 1o | ph } } u. y ) )
15	14	eleq1d 2298	. . . . . . . 8 |- ( x = { { z e. 1o | ph } } -> ( ( x u. y ) e. Fin <-> ( { { z e. 1o | ph } } u. y ) e. Fin ) )
16		uneq2 3352	. . . . . . . . 9 |- ( y = { 1o } -> ( { { z e. 1o | ph } } u. y ) = ( { { z e. 1o | ph } } u. { 1o } ) )
17	16	eleq1d 2298	. . . . . . . 8 |- ( y = { 1o } -> ( ( { { z e. 1o | ph } } u. y ) e. Fin <-> ( { { z e. 1o | ph } } u. { 1o } ) e. Fin ) )
18	15, 17	rspc2v 2920	. . . . . . 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 2302	. . . 4 |- { { z e. 1o | ph } , 1o } e. Fin
22	8	elexi 2812	. . . . 5 |- { z e. 1o | ph } e. _V
23	22	prid1 3772	. . . 4 |- { z e. 1o | ph } e. { { z e. 1o | ph } , 1o }
24	11	elexi 2812	. . . . 5 |- 1o e. _V
25	24	prid2 3773	. . . 4 |- 1o e. { { z e. 1o | ph } , 1o }
26		fidceq 7031	. . . 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 1371	. . 3 |- DECID { z e. 1o | ph } = 1o
28		exmiddc 841	. . 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 2240	. . . 4 |- ( { z e. 1o | ph } = 1o <-> { z e. 1o | ph } = { (/) } )
31		0ex 4211	. . . . 5 |- (/) e. _V
32		biidd 172	. . . . 5 |- ( z = (/) -> ( ph <-> ph ) )
33	31, 32	rabsnt 3741	. . . 4 |- ( { z e. 1o | ph } = { (/) } -> ph )
34	30, 33	sylbi 121	. . 3 |- ( { z e. 1o | ph } = 1o -> ph )
35		df-rab 2517	. . . . 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 2348	. . . . 5 |- ( ph -> 1o = { z | ( z e. 1o /\ ph ) } )
38	35, 37	eqtr4id 2281	. . . 4 |- ( ph -> { z e. 1o | ph } = 1o )
39	38	con3i 635	. . 3 |- ( -. { z e. 1o | ph } = 1o -> -. ph )
40	34, 39	orim12i 764	. 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 713  DECID wdc 839   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   {crab 2512   u. cun 3195   (/)c0 3491   {csn 3666   {cpr 3667   Oncon0 4454   omcom 4682   1oc1o 6555   Fincfn 6887

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1o 6562  df-en 6888  df-fin 6890

This theorem is referenced by: (None)