Theorem unfiexmid 7076

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 3673	. . . . 5 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}, 1o} = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o})
2		unfiexmid.1	. . . . . . 7 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin)
3	2	rgen2a 2584	. . . . . 6 ⊢ ∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin
4		df1o2 6573	. . . . . . . . . 10 ⊢ 1o = {∅}
5		rabeq 2791	. . . . . . . . . 10 ⊢ (1o = {∅} → {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∈ {∅} ∣ 𝜑})
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∈ {∅} ∣ 𝜑}
7		ordtriexmidlem 4610	. . . . . . . . 9 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
8	6, 7	eqeltri 2302	. . . . . . . 8 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ On
9		snfig 6965	. . . . . . . 8 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} ∈ On → {{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin
11		1onn 6664	. . . . . . . 8 ⊢ 1o ∈ ω
12		snfig 6965	. . . . . . . 8 ⊢ (1o ∈ ω → {1o} ∈ Fin)
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ {1o} ∈ Fin
14		uneq1 3351	. . . . . . . . 9 ⊢ (𝑥 = {{𝑧 ∈ 1o ∣ 𝜑}} → (𝑥 ∪ 𝑦) = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦))
15	14	eleq1d 2298	. . . . . . . 8 ⊢ (𝑥 = {{𝑧 ∈ 1o ∣ 𝜑}} → ((𝑥 ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) ∈ Fin))
16		uneq2 3352	. . . . . . . . 9 ⊢ (𝑦 = {1o} → ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}))
17	16	eleq1d 2298	. . . . . . . 8 ⊢ (𝑦 = {1o} → (({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}) ∈ Fin))
18	15, 17	rspc2v 2920	. . . . . . 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 2302	. . . 4 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}, 1o} ∈ Fin
22	8	elexi 2812	. . . . 5 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ V
23	22	prid1 3772	. . . 4 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ {{𝑧 ∈ 1o ∣ 𝜑}, 1o}
24	11	elexi 2812	. . . . 5 ⊢ 1o ∈ V
25	24	prid2 3773	. . . 4 ⊢ 1o ∈ {{𝑧 ∈ 1o ∣ 𝜑}, 1o}
26		fidceq 7027	. . . 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 1371	. . 3 ⊢ DECID {𝑧 ∈ 1o ∣ 𝜑} = 1o
28		exmiddc 841	. . 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 2240	. . . 4 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = 1o ↔ {𝑧 ∈ 1o ∣ 𝜑} = {∅})
31		0ex 4210	. . . . 5 ⊢ ∅ ∈ V
32		biidd 172	. . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
33	31, 32	rabsnt 3741	. . . 4 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = {∅} → 𝜑)
34	30, 33	sylbi 121	. . 3 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = 1o → 𝜑)
35		df-rab 2517	. . . . 5 ⊢ {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∣ (𝑧 ∈ 1o ∧ 𝜑)}
36		iba 300	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ 1o ↔ (𝑧 ∈ 1o ∧ 𝜑)))
37	36	abbi2dv 2348	. . . . 5 ⊢ (𝜑 → 1o = {𝑧 ∣ (𝑧 ∈ 1o ∧ 𝜑)})
38	35, 37	eqtr4id 2281	. . . 4 ⊢ (𝜑 → {𝑧 ∈ 1o ∣ 𝜑} = 1o)
39	38	con3i 635	. . 3 ⊢ (¬ {𝑧 ∈ 1o ∣ 𝜑} = 1o → ¬ 𝜑)
40	34, 39	orim12i 764	. 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 713  DECID wdc 839   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  {crab 2512   ∪ cun 3195  ∅c0 3491  {csn 3666  {cpr 3667  Oncon0 4453  ωcom 4681  1_oc1o 6553  Fincfn 6885

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 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

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 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1o 6560  df-en 6886  df-fin 6888

This theorem is referenced by: (None)