Theorem diffitest 7181

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 4255	. . . . . 6 ⊢ ∅ ∈ V
2		snfig 7093	. . . . . 6 ⊢ (∅ ∈ V → {∅} ∈ Fin)
3	1, 2	ax-mp 5	. . . . 5 ⊢ {∅} ∈ Fin
4		diffitest.1	. . . . 5 ⊢ ∀𝑎 ∈ Fin ∀𝑏(𝑎 ∖ 𝑏) ∈ Fin
5		difeq1 3340	. . . . . . . 8 ⊢ (𝑎 = {∅} → (𝑎 ∖ 𝑏) = ({∅} ∖ 𝑏))
6	5	eleq1d 2307	. . . . . . 7 ⊢ (𝑎 = {∅} → ((𝑎 ∖ 𝑏) ∈ Fin ↔ ({∅} ∖ 𝑏) ∈ Fin))
7	6	albidv 1877	. . . . . 6 ⊢ (𝑎 = {∅} → (∀𝑏(𝑎 ∖ 𝑏) ∈ Fin ↔ ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
8	7	rspcv 2925	. . . . 5 ⊢ ({∅} ∈ Fin → (∀𝑎 ∈ Fin ∀𝑏(𝑎 ∖ 𝑏) ∈ Fin → ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
9	3, 4, 8	mp2 16	. . . 4 ⊢ ∀𝑏({∅} ∖ 𝑏) ∈ Fin
10		rabexg 4274	. . . . . 6 ⊢ ({∅} ∈ Fin → {𝑥 ∈ {∅} ∣ 𝜑} ∈ V)
11	3, 10	ax-mp 5	. . . . 5 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
12		difeq2 3341	. . . . . 6 ⊢ (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → ({∅} ∖ 𝑏) = ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
13	12	eleq1d 2307	. . . . 5 ⊢ (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → (({∅} ∖ 𝑏) ∈ Fin ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin))
14	11, 13	spcv 2919	. . . 4 ⊢ (∀𝑏({∅} ∖ 𝑏) ∈ Fin → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin)
15	9, 14	ax-mp 5	. . 3 ⊢ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin
16		isfi 7037	. . 3 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin ↔ ∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛)
17	15, 16	mpbi 145	. 2 ⊢ ∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛
18		0elnn 4761	. . . . 5 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
19		breq2 4129	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅))
20		en0 7072	. . . . . . . . . 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 3552	. . . . . . . . 9 ⊢ ({𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
24		notrab 3510	. . . . . . . . . 10 ⊢ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = {𝑥 ∈ {∅} ∣ ¬ 𝜑}
25	24	eqeq1i 2246	. . . . . . . . 9 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ {𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅)
26	1	snm 3828	. . . . . . . . . 10 ⊢ ∃𝑤 𝑤 ∈ {∅}
27		r19.3rmv 3615	. . . . . . . . . 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 746	. . . . . 6 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ 𝑛 = ∅) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
32		ensym 7058	. . . . . . . 8 ⊢ (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 → 𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
33		elex2 2838	. . . . . . . 8 ⊢ (∅ ∈ 𝑛 → ∃𝑤 𝑤 ∈ 𝑛)
34		enm 7108	. . . . . . . 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 2982	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (𝑦 ∈ {∅} ∧ ¬ 𝜑))
38	37	simprbi 275	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
39	38	orcd 745	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
40	39, 24	eleq2s 2333	. . . . . . . 8 ⊢ (𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
41	40	exlimiv 1651	. . . . . . 7 ⊢ (∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
42	35, 41	syl 14	. . . . . 6 ⊢ ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
43	31, 42	jaodan 809	. . . . 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 2663	. 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 720  ∀wal 1400   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532  Vcvv 2821   ∖ cdif 3217  ∅c0 3520  {csn 3705   class class class wbr 4125  ωcom 4732   ≈ cen 7010  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-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  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-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-id 4433  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-res 4781  df-ima 4782  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-er 6797  df-en 7013  df-fin 7015

This theorem is referenced by: (None)