Theorem diffitest 6984

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 4171	. . . . . 6 ⊢ ∅ ∈ V
2		snfig 6906	. . . . . 6 ⊢ (∅ ∈ V → {∅} ∈ Fin)
3	1, 2	ax-mp 5	. . . . 5 ⊢ {∅} ∈ Fin
4		diffitest.1	. . . . 5 ⊢ ∀𝑎 ∈ Fin ∀𝑏(𝑎 ∖ 𝑏) ∈ Fin
5		difeq1 3284	. . . . . . . 8 ⊢ (𝑎 = {∅} → (𝑎 ∖ 𝑏) = ({∅} ∖ 𝑏))
6	5	eleq1d 2274	. . . . . . 7 ⊢ (𝑎 = {∅} → ((𝑎 ∖ 𝑏) ∈ Fin ↔ ({∅} ∖ 𝑏) ∈ Fin))
7	6	albidv 1847	. . . . . 6 ⊢ (𝑎 = {∅} → (∀𝑏(𝑎 ∖ 𝑏) ∈ Fin ↔ ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
8	7	rspcv 2873	. . . . 5 ⊢ ({∅} ∈ Fin → (∀𝑎 ∈ Fin ∀𝑏(𝑎 ∖ 𝑏) ∈ Fin → ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
9	3, 4, 8	mp2 16	. . . 4 ⊢ ∀𝑏({∅} ∖ 𝑏) ∈ Fin
10		rabexg 4187	. . . . . 6 ⊢ ({∅} ∈ Fin → {𝑥 ∈ {∅} ∣ 𝜑} ∈ V)
11	3, 10	ax-mp 5	. . . . 5 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
12		difeq2 3285	. . . . . 6 ⊢ (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → ({∅} ∖ 𝑏) = ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
13	12	eleq1d 2274	. . . . 5 ⊢ (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → (({∅} ∖ 𝑏) ∈ Fin ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin))
14	11, 13	spcv 2867	. . . 4 ⊢ (∀𝑏({∅} ∖ 𝑏) ∈ Fin → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin)
15	9, 14	ax-mp 5	. . 3 ⊢ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin
16		isfi 6852	. . 3 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin ↔ ∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛)
17	15, 16	mpbi 145	. 2 ⊢ ∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛
18		0elnn 4667	. . . . 5 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
19		breq2 4048	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅))
20		en0 6887	. . . . . . . . . 10 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅ ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
21	19, 20	bitrdi 196	. . . . . . . . 9 ⊢ (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅))
22	21	biimpac 298	. . . . . . . 8 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ 𝑛 = ∅) → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
23		rabeq0 3490	. . . . . . . . 9 ⊢ ({𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
24		notrab 3450	. . . . . . . . . 10 ⊢ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = {𝑥 ∈ {∅} ∣ ¬ 𝜑}
25	24	eqeq1i 2213	. . . . . . . . 9 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ {𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅)
26	1	snm 3753	. . . . . . . . . 10 ⊢ ∃𝑤 𝑤 ∈ {∅}
27		r19.3rmv 3551	. . . . . . . . . 10 ⊢ (∃𝑤 𝑤 ∈ {∅} → (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑))
28	26, 27	ax-mp 5	. . . . . . . . 9 ⊢ (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
29	23, 25, 28	3bitr4i 212	. . . . . . . 8 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ ¬ ¬ 𝜑)
30	22, 29	sylib 122	. . . . . . 7 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ 𝑛 = ∅) → ¬ ¬ 𝜑)
31	30	olcd 736	. . . . . 6 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ 𝑛 = ∅) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
32		ensym 6873	. . . . . . . 8 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 → 𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
33		elex2 2788	. . . . . . . 8 ⊢ (∅ ∈ 𝑛 → ∃𝑤 𝑤 ∈ 𝑛)
34		enm 6915	. . . . . . . 8 ⊢ ((𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∧ ∃𝑤 𝑤 ∈ 𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
35	32, 33, 34	syl2an 289	. . . . . . 7 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
36		biidd 172	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜑))
37	36	elrab 2929	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (𝑦 ∈ {∅} ∧ ¬ 𝜑))
38	37	simprbi 275	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
39	38	orcd 735	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
40	39, 24	eleq2s 2300	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
41	40	exlimiv 1621	. . . . . . 7 ⊢ (∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
42	35, 41	syl 14	. . . . . 6 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
43	31, 42	jaodan 799	. . . . 5 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ (𝑛 = ∅ ∨ ∅ ∈ 𝑛)) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
44	18, 43	sylan2 286	. . . 4 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ 𝑛 ∈ ω) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
45	44	ancoms 268	. . 3 ⊢ ((𝑛 ∈ ω ∧ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
46	45	rexlimiva 2618	. 2 ⊢ (∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
47	17, 46	ax-mp 5	1 ⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 710  ∀wal 1371   = wceq 1373  ∃wex 1515   ∈ wcel 2176  ∀wral 2484  ∃wrex 2485  {crab 2488  Vcvv 2772   ∖ cdif 3163  ∅c0 3460  {csn 3633   class class class wbr 4044  ωcom 4638   ≈ cen 6825  Fincfn 6827

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-id 4340  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6502  df-er 6620  df-en 6828  df-fin 6830

This theorem is referenced by: (None)