Theorem unfiexmid 6931

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 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin)

Assertion

Ref	Expression
unfiexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem unfiexmid

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pr 3611	. . . . 5 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}, 1o} = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o})
2		unfiexmid.1	. . . . . . 7 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin)
3	2	rgen2a 2541	. . . . . 6 ⊢ ∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin
4		df1o2 6444	. . . . . . . . . 10 ⊢ 1o = {∅}
5		rabeq 2741	. . . . . . . . . 10 ⊢ (1o = {∅} → {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∈ {∅} ∣ 𝜑})
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∈ {∅} ∣ 𝜑}
7		ordtriexmidlem 4530	. . . . . . . . 9 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
8	6, 7	eqeltri 2260	. . . . . . . 8 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ On
9		snfig 6828	. . . . . . . 8 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} ∈ On → {{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin
11		1onn 6535	. . . . . . . 8 ⊢ 1o ∈ ω
12		snfig 6828	. . . . . . . 8 ⊢ (1o ∈ ω → {1o} ∈ Fin)
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ {1o} ∈ Fin
14		uneq1 3294	. . . . . . . . 9 ⊢ (𝑥 = {{𝑧 ∈ 1o ∣ 𝜑}} → (𝑥 ∪ 𝑦) = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦))
15	14	eleq1d 2256	. . . . . . . 8 ⊢ (𝑥 = {{𝑧 ∈ 1o ∣ 𝜑}} → ((𝑥 ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) ∈ Fin))
16		uneq2 3295	. . . . . . . . 9 ⊢ (𝑦 = {1o} → ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}))
17	16	eleq1d 2256	. . . . . . . 8 ⊢ (𝑦 = {1o} → (({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}) ∈ Fin))
18	15, 17	rspc2v 2866	. . . . . . 7 ⊢ (({{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin ∧ {1o} ∈ Fin) → (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin → ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}) ∈ Fin))
19	10, 13, 18	mp2an 426	. . . . . 6 ⊢ (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin → ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}) ∈ Fin)
20	3, 19	ax-mp 5	. . . . 5 ⊢ ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}) ∈ Fin
21	1, 20	eqeltri 2260	. . . 4 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}, 1o} ∈ Fin
22	8	elexi 2761	. . . . 5 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ V
23	22	prid1 3710	. . . 4 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ {{𝑧 ∈ 1o ∣ 𝜑}, 1o}
24	11	elexi 2761	. . . . 5 ⊢ 1o ∈ V
25	24	prid2 3711	. . . 4 ⊢ 1o ∈ {{𝑧 ∈ 1o ∣ 𝜑}, 1o}
26		fidceq 6883	. . . 4 ⊢ (({{𝑧 ∈ 1o ∣ 𝜑}, 1o} ∈ Fin ∧ {𝑧 ∈ 1o ∣ 𝜑} ∈ {{𝑧 ∈ 1o ∣ 𝜑}, 1o} ∧ 1o ∈ {{𝑧 ∈ 1o ∣ 𝜑}, 1o}) → DECID {𝑧 ∈ 1o ∣ 𝜑} = 1o)
27	21, 23, 25, 26	mp3an 1347	. . 3 ⊢ DECID {𝑧 ∈ 1o ∣ 𝜑} = 1o
28		exmiddc 837	. . 3 ⊢ (DECID {𝑧 ∈ 1o ∣ 𝜑} = 1o → ({𝑧 ∈ 1o ∣ 𝜑} = 1o ∨ ¬ {𝑧 ∈ 1o ∣ 𝜑} = 1o))
29	27, 28	ax-mp 5	. 2 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = 1o ∨ ¬ {𝑧 ∈ 1o ∣ 𝜑} = 1o)
30	4	eqeq2i 2198	. . . 4 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = 1o ↔ {𝑧 ∈ 1o ∣ 𝜑} = {∅})
31		0ex 4142	. . . . 5 ⊢ ∅ ∈ V
32		biidd 172	. . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
33	31, 32	rabsnt 3679	. . . 4 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = {∅} → 𝜑)
34	30, 33	sylbi 121	. . 3 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = 1o → 𝜑)
35		df-rab 2474	. . . . 5 ⊢ {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∣ (𝑧 ∈ 1o ∧ 𝜑)}
36		iba 300	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ 1o ↔ (𝑧 ∈ 1o ∧ 𝜑)))
37	36	abbi2dv 2306	. . . . 5 ⊢ (𝜑 → 1o = {𝑧 ∣ (𝑧 ∈ 1o ∧ 𝜑)})
38	35, 37	eqtr4id 2239	. . . 4 ⊢ (𝜑 → {𝑧 ∈ 1o ∣ 𝜑} = 1o)
39	38	con3i 633	. . 3 ⊢ (¬ {𝑧 ∈ 1o ∣ 𝜑} = 1o → ¬ 𝜑)
40	34, 39	orim12i 760	. 2 ⊢ (({𝑧 ∈ 1o ∣ 𝜑} = 1o ∨ ¬ {𝑧 ∈ 1o ∣ 𝜑} = 1o) → (𝜑 ∨ ¬ 𝜑))
41	29, 40	ax-mp 5	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   = wceq 1363   ∈ wcel 2158  {cab 2173  ∀wral 2465  {crab 2469   ∪ cun 3139  ∅c0 3434  {csn 3604  {cpr 3605  Oncon0 4375  ωcom 4601  1_oc1o 6424  Fincfn 6754

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1o 6431  df-en 6755  df-fin 6757

This theorem is referenced by: (None)