Theorem unfiexmid 6735

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 3481	. . . . 5 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}, 1o} = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o})
2		unfiexmid.1	. . . . . . 7 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin)
3	2	rgen2a 2445	. . . . . 6 ⊢ ∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin
4		df1o2 6256	. . . . . . . . . 10 ⊢ 1o = {∅}
5		rabeq 2633	. . . . . . . . . 10 ⊢ (1o = {∅} → {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∈ {∅} ∣ 𝜑})
6	4, 5	ax-mp 7	. . . . . . . . 9 ⊢ {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∈ {∅} ∣ 𝜑}
7		ordtriexmidlem 4373	. . . . . . . . 9 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
8	6, 7	eqeltri 2172	. . . . . . . 8 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ On
9		snfig 6638	. . . . . . . 8 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} ∈ On → {{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin)
10	8, 9	ax-mp 7	. . . . . . 7 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin
11		1onn 6346	. . . . . . . 8 ⊢ 1o ∈ ω
12		snfig 6638	. . . . . . . 8 ⊢ (1o ∈ ω → {1o} ∈ Fin)
13	11, 12	ax-mp 7	. . . . . . 7 ⊢ {1o} ∈ Fin
14		uneq1 3170	. . . . . . . . 9 ⊢ (𝑥 = {{𝑧 ∈ 1o ∣ 𝜑}} → (𝑥 ∪ 𝑦) = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦))
15	14	eleq1d 2168	. . . . . . . 8 ⊢ (𝑥 = {{𝑧 ∈ 1o ∣ 𝜑}} → ((𝑥 ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) ∈ Fin))
16		uneq2 3171	. . . . . . . . 9 ⊢ (𝑦 = {1o} → ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}))
17	16	eleq1d 2168	. . . . . . . 8 ⊢ (𝑦 = {1o} → (({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}) ∈ Fin))
18	15, 17	rspc2v 2756	. . . . . . 7 ⊢ (({{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin ∧ {1o} ∈ Fin) → (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin → ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}) ∈ Fin))
19	10, 13, 18	mp2an 420	. . . . . 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 2172	. . . 4 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}, 1o} ∈ Fin
22	8	elexi 2653	. . . . 5 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ V
23	22	prid1 3576	. . . 4 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ {{𝑧 ∈ 1o ∣ 𝜑}, 1o}
24	11	elexi 2653	. . . . 5 ⊢ 1o ∈ V
25	24	prid2 3577	. . . 4 ⊢ 1o ∈ {{𝑧 ∈ 1o ∣ 𝜑}, 1o}
26		fidceq 6692	. . . 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 1283	. . 3 ⊢ DECID {𝑧 ∈ 1o ∣ 𝜑} = 1o
28		exmiddc 788	. . 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 2110	. . . 4 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = 1o ↔ {𝑧 ∈ 1o ∣ 𝜑} = {∅})
31		0ex 3995	. . . . 5 ⊢ ∅ ∈ V
32		biidd 171	. . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
33	31, 32	rabsnt 3545	. . . 4 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = {∅} → 𝜑)
34	30, 33	sylbi 120	. . 3 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = 1o → 𝜑)
35		iba 296	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ 1o ↔ (𝑧 ∈ 1o ∧ 𝜑)))
36	35	abbi2dv 2218	. . . . 5 ⊢ (𝜑 → 1o = {𝑧 ∣ (𝑧 ∈ 1o ∧ 𝜑)})
37		df-rab 2384	. . . . 5 ⊢ {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∣ (𝑧 ∈ 1o ∧ 𝜑)}
38	36, 37	syl6reqr 2151	. . . 4 ⊢ (𝜑 → {𝑧 ∈ 1o ∣ 𝜑} = 1o)
39	38	con3i 602	. . 3 ⊢ (¬ {𝑧 ∈ 1o ∣ 𝜑} = 1o → ¬ 𝜑)
40	34, 39	orim12i 717	. 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 103   ∨ wo 670  DECID wdc 786   = wceq 1299   ∈ wcel 1448  {cab 2086  ∀wral 2375  {crab 2379   ∪ cun 3019  ∅c0 3310  {csn 3474  {cpr 3475  Oncon0 4223  ωcom 4442  1_oc1o 6236  Fincfn 6564

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-1o 6243  df-en 6565  df-fin 6567

This theorem is referenced by: (None)