Theorem unfiexmid 6462

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 3423	. . . . 5 ⊢ {{𝑧 ∈ 1𝑜 ∣ 𝜑}, 1𝑜} = ({{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∪ {1𝑜})
2		unfiexmid.1	. . . . . . 7 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin)
3	2	rgen2a 2422	. . . . . 6 ⊢ ∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin
4		df1o2 6097	. . . . . . . . . 10 ⊢ 1𝑜 = {∅}
5		rabeq 2602	. . . . . . . . . 10 ⊢ (1𝑜 = {∅} → {𝑧 ∈ 1𝑜 ∣ 𝜑} = {𝑧 ∈ {∅} ∣ 𝜑})
6	4, 5	ax-mp 7	. . . . . . . . 9 ⊢ {𝑧 ∈ 1𝑜 ∣ 𝜑} = {𝑧 ∈ {∅} ∣ 𝜑}
7		ordtriexmidlem 4291	. . . . . . . . 9 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
8	6, 7	eqeltri 2155	. . . . . . . 8 ⊢ {𝑧 ∈ 1𝑜 ∣ 𝜑} ∈ On
9		snfig 6380	. . . . . . . 8 ⊢ ({𝑧 ∈ 1𝑜 ∣ 𝜑} ∈ On → {{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∈ Fin)
10	8, 9	ax-mp 7	. . . . . . 7 ⊢ {{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∈ Fin
11		1onn 6180	. . . . . . . 8 ⊢ 1𝑜 ∈ ω
12		snfig 6380	. . . . . . . 8 ⊢ (1𝑜 ∈ ω → {1𝑜} ∈ Fin)
13	11, 12	ax-mp 7	. . . . . . 7 ⊢ {1𝑜} ∈ Fin
14		uneq1 3129	. . . . . . . . 9 ⊢ (𝑥 = {{𝑧 ∈ 1𝑜 ∣ 𝜑}} → (𝑥 ∪ 𝑦) = ({{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∪ 𝑦))
15	14	eleq1d 2151	. . . . . . . 8 ⊢ (𝑥 = {{𝑧 ∈ 1𝑜 ∣ 𝜑}} → ((𝑥 ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∪ 𝑦) ∈ Fin))
16		uneq2 3130	. . . . . . . . 9 ⊢ (𝑦 = {1𝑜} → ({{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∪ 𝑦) = ({{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∪ {1𝑜}))
17	16	eleq1d 2151	. . . . . . . 8 ⊢ (𝑦 = {1𝑜} → (({{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∪ {1𝑜}) ∈ Fin))
18	15, 17	rspc2v 2721	. . . . . . 7 ⊢ (({{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∈ Fin ∧ {1𝑜} ∈ Fin) → (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin → ({{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∪ {1𝑜}) ∈ Fin))
19	10, 13, 18	mp2an 417	. . . . . 6 ⊢ (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin → ({{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∪ {1𝑜}) ∈ Fin)
20	3, 19	ax-mp 7	. . . . 5 ⊢ ({{𝑧 ∈ 1𝑜 ∣ 𝜑}} ∪ {1𝑜}) ∈ Fin
21	1, 20	eqeltri 2155	. . . 4 ⊢ {{𝑧 ∈ 1𝑜 ∣ 𝜑}, 1𝑜} ∈ Fin
22	8	elexi 2620	. . . . 5 ⊢ {𝑧 ∈ 1𝑜 ∣ 𝜑} ∈ V
23	22	prid1 3516	. . . 4 ⊢ {𝑧 ∈ 1𝑜 ∣ 𝜑} ∈ {{𝑧 ∈ 1𝑜 ∣ 𝜑}, 1𝑜}
24	11	elexi 2620	. . . . 5 ⊢ 1𝑜 ∈ V
25	24	prid2 3517	. . . 4 ⊢ 1𝑜 ∈ {{𝑧 ∈ 1𝑜 ∣ 𝜑}, 1𝑜}
26		fidceq 6425	. . . 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 1269	. . 3 ⊢ DECID {𝑧 ∈ 1𝑜 ∣ 𝜑} = 1𝑜
28		exmiddc 778	. . 3 ⊢ (DECID {𝑧 ∈ 1𝑜 ∣ 𝜑} = 1𝑜 → ({𝑧 ∈ 1𝑜 ∣ 𝜑} = 1𝑜 ∨ ¬ {𝑧 ∈ 1𝑜 ∣ 𝜑} = 1𝑜))
29	27, 28	ax-mp 7	. 2 ⊢ ({𝑧 ∈ 1𝑜 ∣ 𝜑} = 1𝑜 ∨ ¬ {𝑧 ∈ 1𝑜 ∣ 𝜑} = 1𝑜)
30	4	eqeq2i 2093	. . . 4 ⊢ ({𝑧 ∈ 1𝑜 ∣ 𝜑} = 1𝑜 ↔ {𝑧 ∈ 1𝑜 ∣ 𝜑} = {∅})
31		0ex 3925	. . . . 5 ⊢ ∅ ∈ V
32		biidd 170	. . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
33	31, 32	rabsnt 3485	. . . 4 ⊢ ({𝑧 ∈ 1𝑜 ∣ 𝜑} = {∅} → 𝜑)
34	30, 33	sylbi 119	. . 3 ⊢ ({𝑧 ∈ 1𝑜 ∣ 𝜑} = 1𝑜 → 𝜑)
35		iba 294	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ 1𝑜 ↔ (𝑧 ∈ 1𝑜 ∧ 𝜑)))
36	35	abbi2dv 2201	. . . . 5 ⊢ (𝜑 → 1𝑜 = {𝑧 ∣ (𝑧 ∈ 1𝑜 ∧ 𝜑)})
37		df-rab 2362	. . . . 5 ⊢ {𝑧 ∈ 1𝑜 ∣ 𝜑} = {𝑧 ∣ (𝑧 ∈ 1𝑜 ∧ 𝜑)}
38	36, 37	syl6reqr 2134	. . . 4 ⊢ (𝜑 → {𝑧 ∈ 1𝑜 ∣ 𝜑} = 1𝑜)
39	38	con3i 595	. . 3 ⊢ (¬ {𝑧 ∈ 1𝑜 ∣ 𝜑} = 1𝑜 → ¬ 𝜑)
40	34, 39	orim12i 709	. 2 ⊢ (({𝑧 ∈ 1𝑜 ∣ 𝜑} = 1𝑜 ∨ ¬ {𝑧 ∈ 1𝑜 ∣ 𝜑} = 1𝑜) → (𝜑 ∨ ¬ 𝜑))
41	29, 40	ax-mp 7	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∨ wo 662  DECID wdc 776   = wceq 1285   ∈ wcel 1434  {cab 2069  ∀wral 2353  {crab 2357   ∪ cun 2980  ∅c0 3267  {csn 3416  {cpr 3417  Oncon0 4146  ωcom 4359  1_𝑜c1o 6078  Fincfn 6308

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-1o 6085  df-en 6309  df-fin 6311

This theorem is referenced by: (None)