Theorem diffitest 6781

Description: If subtracting any set from a finite set gives a finite set, any proposition of the form ¬ 𝜑 is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove 𝐴 ∈ Fin → (𝐴 ∖ 𝐵) ∈ Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)

Hypothesis

Ref	Expression
diffitest.1	⊢ ∀𝑎 ∈ Fin ∀𝑏(𝑎 ∖ 𝑏) ∈ Fin

Assertion

Ref	Expression
diffitest	⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑)

Distinct variable groups:   𝑎,𝑏   𝜑,𝑏

Allowed substitution hint:   𝜑(𝑎)

Proof of Theorem diffitest

Dummy variables 𝑥 𝑛 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4055	. . . . . 6 ⊢ ∅ ∈ V
2		snfig 6708	. . . . . 6 ⊢ (∅ ∈ V → {∅} ∈ Fin)
3	1, 2	ax-mp 5	. . . . 5 ⊢ {∅} ∈ Fin
4		diffitest.1	. . . . 5 ⊢ ∀𝑎 ∈ Fin ∀𝑏(𝑎 ∖ 𝑏) ∈ Fin
5		difeq1 3187	. . . . . . . 8 ⊢ (𝑎 = {∅} → (𝑎 ∖ 𝑏) = ({∅} ∖ 𝑏))
6	5	eleq1d 2208	. . . . . . 7 ⊢ (𝑎 = {∅} → ((𝑎 ∖ 𝑏) ∈ Fin ↔ ({∅} ∖ 𝑏) ∈ Fin))
7	6	albidv 1796	. . . . . 6 ⊢ (𝑎 = {∅} → (∀𝑏(𝑎 ∖ 𝑏) ∈ Fin ↔ ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
8	7	rspcv 2785	. . . . 5 ⊢ ({∅} ∈ Fin → (∀𝑎 ∈ Fin ∀𝑏(𝑎 ∖ 𝑏) ∈ Fin → ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
9	3, 4, 8	mp2 16	. . . 4 ⊢ ∀𝑏({∅} ∖ 𝑏) ∈ Fin
10		rabexg 4071	. . . . . 6 ⊢ ({∅} ∈ Fin → {𝑥 ∈ {∅} ∣ 𝜑} ∈ V)
11	3, 10	ax-mp 5	. . . . 5 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
12		difeq2 3188	. . . . . 6 ⊢ (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → ({∅} ∖ 𝑏) = ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
13	12	eleq1d 2208	. . . . 5 ⊢ (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → (({∅} ∖ 𝑏) ∈ Fin ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin))
14	11, 13	spcv 2779	. . . 4 ⊢ (∀𝑏({∅} ∖ 𝑏) ∈ Fin → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin)
15	9, 14	ax-mp 5	. . 3 ⊢ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin
16		isfi 6655	. . 3 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin ↔ ∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛)
17	15, 16	mpbi 144	. 2 ⊢ ∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛
18		0elnn 4532	. . . . 5 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
19		breq2 3933	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅))
20		en0 6689	. . . . . . . . . 10 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅ ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
21	19, 20	syl6bb 195	. . . . . . . . 9 ⊢ (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅))
22	21	biimpac 296	. . . . . . . 8 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ 𝑛 = ∅) → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
23		rabeq0 3392	. . . . . . . . 9 ⊢ ({𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
24		notrab 3353	. . . . . . . . . 10 ⊢ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = {𝑥 ∈ {∅} ∣ ¬ 𝜑}
25	24	eqeq1i 2147	. . . . . . . . 9 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ {𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅)
26	1	snm 3643	. . . . . . . . . 10 ⊢ ∃𝑤 𝑤 ∈ {∅}
27		r19.3rmv 3453	. . . . . . . . . 10 ⊢ (∃𝑤 𝑤 ∈ {∅} → (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑))
28	26, 27	ax-mp 5	. . . . . . . . 9 ⊢ (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
29	23, 25, 28	3bitr4i 211	. . . . . . . 8 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ ¬ ¬ 𝜑)
30	22, 29	sylib 121	. . . . . . 7 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ 𝑛 = ∅) → ¬ ¬ 𝜑)
31	30	olcd 723	. . . . . 6 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ 𝑛 = ∅) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
32		ensym 6675	. . . . . . . 8 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 → 𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
33		elex2 2702	. . . . . . . 8 ⊢ (∅ ∈ 𝑛 → ∃𝑤 𝑤 ∈ 𝑛)
34		enm 6714	. . . . . . . 8 ⊢ ((𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∧ ∃𝑤 𝑤 ∈ 𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
35	32, 33, 34	syl2an 287	. . . . . . 7 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
36		biidd 171	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜑))
37	36	elrab 2840	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (𝑦 ∈ {∅} ∧ ¬ 𝜑))
38	37	simprbi 273	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
39	38	orcd 722	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
40	39, 24	eleq2s 2234	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
41	40	exlimiv 1577	. . . . . . 7 ⊢ (∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
42	35, 41	syl 14	. . . . . 6 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
43	31, 42	jaodan 786	. . . . 5 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ (𝑛 = ∅ ∨ ∅ ∈ 𝑛)) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
44	18, 43	sylan2 284	. . . 4 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ 𝑛 ∈ ω) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
45	44	ancoms 266	. . 3 ⊢ ((𝑛 ∈ ω ∧ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
46	45	rexlimiva 2544	. 2 ⊢ (∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
47	17, 46	ax-mp 5	1 ⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104   ∨ wo 697  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420  Vcvv 2686   ∖ cdif 3068  ∅c0 3363  {csn 3527   class class class wbr 3929  ωcom 4504   ≈ cen 6632  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-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  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-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-id 4215  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-res 4551  df-ima 4552  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-er 6429  df-en 6635  df-fin 6637

This theorem is referenced by: (None)