Theorem unfiexmid 6806

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 3534	. . . . 5 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}, 1o} = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o})
2		unfiexmid.1	. . . . . . 7 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin)
3	2	rgen2a 2486	. . . . . 6 ⊢ ∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin
4		df1o2 6326	. . . . . . . . . 10 ⊢ 1o = {∅}
5		rabeq 2678	. . . . . . . . . 10 ⊢ (1o = {∅} → {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∈ {∅} ∣ 𝜑})
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∈ {∅} ∣ 𝜑}
7		ordtriexmidlem 4435	. . . . . . . . 9 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
8	6, 7	eqeltri 2212	. . . . . . . 8 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ On
9		snfig 6708	. . . . . . . 8 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} ∈ On → {{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin
11		1onn 6416	. . . . . . . 8 ⊢ 1o ∈ ω
12		snfig 6708	. . . . . . . 8 ⊢ (1o ∈ ω → {1o} ∈ Fin)
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ {1o} ∈ Fin
14		uneq1 3223	. . . . . . . . 9 ⊢ (𝑥 = {{𝑧 ∈ 1o ∣ 𝜑}} → (𝑥 ∪ 𝑦) = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦))
15	14	eleq1d 2208	. . . . . . . 8 ⊢ (𝑥 = {{𝑧 ∈ 1o ∣ 𝜑}} → ((𝑥 ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) ∈ Fin))
16		uneq2 3224	. . . . . . . . 9 ⊢ (𝑦 = {1o} → ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}))
17	16	eleq1d 2208	. . . . . . . 8 ⊢ (𝑦 = {1o} → (({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}) ∈ Fin))
18	15, 17	rspc2v 2802	. . . . . . 7 ⊢ (({{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin ∧ {1o} ∈ Fin) → (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin → ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}) ∈ Fin))
19	10, 13, 18	mp2an 422	. . . . . 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 2212	. . . 4 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}, 1o} ∈ Fin
22	8	elexi 2698	. . . . 5 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ V
23	22	prid1 3629	. . . 4 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ {{𝑧 ∈ 1o ∣ 𝜑}, 1o}
24	11	elexi 2698	. . . . 5 ⊢ 1o ∈ V
25	24	prid2 3630	. . . 4 ⊢ 1o ∈ {{𝑧 ∈ 1o ∣ 𝜑}, 1o}
26		fidceq 6763	. . . 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 1315	. . 3 ⊢ DECID {𝑧 ∈ 1o ∣ 𝜑} = 1o
28		exmiddc 821	. . 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 2150	. . . 4 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = 1o ↔ {𝑧 ∈ 1o ∣ 𝜑} = {∅})
31		0ex 4055	. . . . 5 ⊢ ∅ ∈ V
32		biidd 171	. . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
33	31, 32	rabsnt 3598	. . . 4 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = {∅} → 𝜑)
34	30, 33	sylbi 120	. . 3 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = 1o → 𝜑)
35		iba 298	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ 1o ↔ (𝑧 ∈ 1o ∧ 𝜑)))
36	35	abbi2dv 2258	. . . . 5 ⊢ (𝜑 → 1o = {𝑧 ∣ (𝑧 ∈ 1o ∧ 𝜑)})
37		df-rab 2425	. . . . 5 ⊢ {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∣ (𝑧 ∈ 1o ∧ 𝜑)}
38	36, 37	syl6reqr 2191	. . . 4 ⊢ (𝜑 → {𝑧 ∈ 1o ∣ 𝜑} = 1o)
39	38	con3i 621	. . 3 ⊢ (¬ {𝑧 ∈ 1o ∣ 𝜑} = 1o → ¬ 𝜑)
40	34, 39	orim12i 748	. 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 103   ∨ wo 697  DECID wdc 819   = wceq 1331   ∈ wcel 1480  {cab 2125  ∀wral 2416  {crab 2420   ∪ cun 3069  ∅c0 3363  {csn 3527  {cpr 3528  Oncon0 4285  ωcom 4504  1_oc1o 6306  Fincfn 6634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1o 6313  df-en 6635  df-fin 6637

This theorem is referenced by: (None)