Theorem unfiexmid 7215

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 3712	. . . . 5 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}, 1o} = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o})
2		unfiexmid.1	. . . . . . 7 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin)
3	2	rgen2a 2604	. . . . . 6 ⊢ ∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin
4		df1o2 6691	. . . . . . . . . 10 ⊢ 1o = {∅}
5		rabeq 2813	. . . . . . . . . 10 ⊢ (1o = {∅} → {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∈ {∅} ∣ 𝜑})
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∈ {∅} ∣ 𝜑}
7		ordtriexmidlem 4661	. . . . . . . . 9 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
8	6, 7	eqeltri 2311	. . . . . . . 8 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ On
9		snfig 7093	. . . . . . . 8 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} ∈ On → {{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin
11		1onn 6783	. . . . . . . 8 ⊢ 1o ∈ ω
12		snfig 7093	. . . . . . . 8 ⊢ (1o ∈ ω → {1o} ∈ Fin)
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ {1o} ∈ Fin
14		uneq1 3376	. . . . . . . . 9 ⊢ (𝑥 = {{𝑧 ∈ 1o ∣ 𝜑}} → (𝑥 ∪ 𝑦) = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦))
15	14	eleq1d 2307	. . . . . . . 8 ⊢ (𝑥 = {{𝑧 ∈ 1o ∣ 𝜑}} → ((𝑥 ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) ∈ Fin))
16		uneq2 3377	. . . . . . . . 9 ⊢ (𝑦 = {1o} → ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) = ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}))
17	16	eleq1d 2307	. . . . . . . 8 ⊢ (𝑦 = {1o} → (({{𝑧 ∈ 1o ∣ 𝜑}} ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}) ∈ Fin))
18	15, 17	rspc2v 2943	. . . . . . 7 ⊢ (({{𝑧 ∈ 1o ∣ 𝜑}} ∈ Fin ∧ {1o} ∈ Fin) → (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥 ∪ 𝑦) ∈ Fin → ({{𝑧 ∈ 1o ∣ 𝜑}} ∪ {1o}) ∈ Fin))
19	10, 13, 18	mp2an 430	. . . . . 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 2311	. . . 4 ⊢ {{𝑧 ∈ 1o ∣ 𝜑}, 1o} ∈ Fin
22	8	elexi 2834	. . . . 5 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ V
23	22	prid1 3813	. . . 4 ⊢ {𝑧 ∈ 1o ∣ 𝜑} ∈ {{𝑧 ∈ 1o ∣ 𝜑}, 1o}
24	11	elexi 2834	. . . . 5 ⊢ 1o ∈ V
25	24	prid2 3814	. . . 4 ⊢ 1o ∈ {{𝑧 ∈ 1o ∣ 𝜑}, 1o}
26		fidceq 7161	. . . 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 1378	. . 3 ⊢ DECID {𝑧 ∈ 1o ∣ 𝜑} = 1o
28		exmiddc 848	. . 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 2249	. . . 4 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = 1o ↔ {𝑧 ∈ 1o ∣ 𝜑} = {∅})
31		0ex 4255	. . . . 5 ⊢ ∅ ∈ V
32		biidd 172	. . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
33	31, 32	rabsnt 3782	. . . 4 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = {∅} → 𝜑)
34	30, 33	sylbi 121	. . 3 ⊢ ({𝑧 ∈ 1o ∣ 𝜑} = 1o → 𝜑)
35		df-rab 2537	. . . . 5 ⊢ {𝑧 ∈ 1o ∣ 𝜑} = {𝑧 ∣ (𝑧 ∈ 1o ∧ 𝜑)}
36		iba 300	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ 1o ↔ (𝑧 ∈ 1o ∧ 𝜑)))
37	36	abbi2dv 2359	. . . . 5 ⊢ (𝜑 → 1o = {𝑧 ∣ (𝑧 ∈ 1o ∧ 𝜑)})
38	35, 37	eqtr4id 2290	. . . 4 ⊢ (𝜑 → {𝑧 ∈ 1o ∣ 𝜑} = 1o)
39	38	con3i 641	. . 3 ⊢ (¬ {𝑧 ∈ 1o ∣ 𝜑} = 1o → ¬ 𝜑)
40	34, 39	orim12i 771	. 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 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  {crab 2532   ∪ cun 3218  ∅c0 3520  {csn 3705  {cpr 3706  Oncon0 4503  ωcom 4732  1_oc1o 6670  Fincfn 7012

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1o 6677  df-en 7013  df-fin 7015

This theorem is referenced by: (None)