Theorem wexmiddifxylem 17250

Description: Lemma for wexmiddifxylem 17250. Showing weak excluded middle given a suitable finite set. (Contributed by Jim Kingdon, 1-Aug-2026.)

Assertion

Ref	Expression
wexmiddifxylem	⊢ (({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) ∈ Fin → DECID ¬ 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem wexmiddifxylem

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin0or 7194	. . . 4 ⊢ (({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) ∈ Fin → (({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) = ∅ ∨ ∃𝑗 𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}})))
2		notm0 3542	. . . . . 6 ⊢ (¬ ∃𝑗 𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) ↔ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) = ∅)
3		1oex 6699	. . . . . . . . 9 ⊢ 1o ∈ V
4	3	snm 3833	. . . . . . . 8 ⊢ ∃𝑗 𝑗 ∈ {1o}
5		0ex 4262	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
6	5	snm 3833	. . . . . . . . . . . . . . . 16 ⊢ ∃𝑤 𝑤 ∈ {∅}
7		df1o2 6705	. . . . . . . . . . . . . . . . . 18 ⊢ 1o = {∅}
8	7	eleq2i 2305	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ 1o ↔ 𝑤 ∈ {∅})
9	8	exbii 1658	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 𝑤 ∈ 1o ↔ ∃𝑤 𝑤 ∈ {∅})
10	6, 9	mpbir 146	. . . . . . . . . . . . . . 15 ⊢ ∃𝑤 𝑤 ∈ 1o
11		r19.3rmv 3618	. . . . . . . . . . . . . . 15 ⊢ (∃𝑤 𝑤 ∈ 1o → (¬ 𝜑 ↔ ∀𝑥 ∈ 1o ¬ 𝜑))
12	10, 11	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (¬ 𝜑 ↔ ∀𝑥 ∈ 1o ¬ 𝜑)
13		rabeq0 3552	. . . . . . . . . . . . . 14 ⊢ ({𝑥 ∈ 1o ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ 1o ¬ 𝜑)
14	12, 13	sylbb2 138	. . . . . . . . . . . . 13 ⊢ (¬ 𝜑 → {𝑥 ∈ 1o ∣ 𝜑} = ∅)
15	14	sneqd 3722	. . . . . . . . . . . 12 ⊢ (¬ 𝜑 → {{𝑥 ∈ 1o ∣ 𝜑}} = {∅})
16	15	difeq2d 3347	. . . . . . . . . . 11 ⊢ (¬ 𝜑 → ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) = ({1o} ∖ {∅}))
17		1n0 6709	. . . . . . . . . . . . . 14 ⊢ 1o ≠ ∅
18	17	nesymi 2466	. . . . . . . . . . . . 13 ⊢ ¬ ∅ = 1o
19		elsni 3727	. . . . . . . . . . . . 13 ⊢ (∅ ∈ {1o} → ∅ = 1o)
20	18, 19	mto 672	. . . . . . . . . . . 12 ⊢ ¬ ∅ ∈ {1o}
21		difsn 3852	. . . . . . . . . . . 12 ⊢ (¬ ∅ ∈ {1o} → ({1o} ∖ {∅}) = {1o})
22	20, 21	ax-mp 5	. . . . . . . . . . 11 ⊢ ({1o} ∖ {∅}) = {1o}
23	16, 22	eqtrdi 2287	. . . . . . . . . 10 ⊢ (¬ 𝜑 → ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) = {1o})
24	23	eleq2d 2308	. . . . . . . . 9 ⊢ (¬ 𝜑 → (𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) ↔ 𝑗 ∈ {1o}))
25	24	exbidv 1878	. . . . . . . 8 ⊢ (¬ 𝜑 → (∃𝑗 𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) ↔ ∃𝑗 𝑗 ∈ {1o}))
26	4, 25	mpbiri 168	. . . . . . 7 ⊢ (¬ 𝜑 → ∃𝑗 𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}))
27	26	con3i 641	. . . . . 6 ⊢ (¬ ∃𝑗 𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) → ¬ ¬ 𝜑)
28	2, 27	sylbir 135	. . . . 5 ⊢ (({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) = ∅ → ¬ ¬ 𝜑)
29		eldifn 3352	. . . . . . . . . 10 ⊢ (𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) → ¬ 𝑗 ∈ {{𝑥 ∈ 1o ∣ 𝜑}})
30		velsn 3726	. . . . . . . . . 10 ⊢ (𝑗 ∈ {{𝑥 ∈ 1o ∣ 𝜑}} ↔ 𝑗 = {𝑥 ∈ 1o ∣ 𝜑})
31	29, 30	sylnib 687	. . . . . . . . 9 ⊢ (𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) → ¬ 𝑗 = {𝑥 ∈ 1o ∣ 𝜑})
32		eldifi 3351	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) → 𝑗 ∈ {1o})
33		elsni 3727	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {1o} → 𝑗 = 1o)
34	32, 33	syl 14	. . . . . . . . . 10 ⊢ (𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) → 𝑗 = 1o)
35	34	eqeq1d 2247	. . . . . . . . 9 ⊢ (𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) → (𝑗 = {𝑥 ∈ 1o ∣ 𝜑} ↔ 1o = {𝑥 ∈ 1o ∣ 𝜑}))
36	31, 35	mtbid 683	. . . . . . . 8 ⊢ (𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) → ¬ 1o = {𝑥 ∈ 1o ∣ 𝜑})
37	36	neqcomd 2243	. . . . . . 7 ⊢ (𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) → ¬ {𝑥 ∈ 1o ∣ 𝜑} = 1o)
38		rabid1o 17239	. . . . . . 7 ⊢ ({𝑥 ∈ 1o ∣ 𝜑} = 1o ↔ 𝜑)
39	37, 38	sylnib 687	. . . . . 6 ⊢ (𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) → ¬ 𝜑)
40	39	exlimiv 1651	. . . . 5 ⊢ (∃𝑗 𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) → ¬ 𝜑)
41	28, 40	orim12i 771	. . . 4 ⊢ ((({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) = ∅ ∨ ∃𝑗 𝑗 ∈ ({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}})) → (¬ ¬ 𝜑 ∨ ¬ 𝜑))
42	1, 41	syl 14	. . 3 ⊢ (({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) ∈ Fin → (¬ ¬ 𝜑 ∨ ¬ 𝜑))
43	42	orcomd 741	. 2 ⊢ (({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) ∈ Fin → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
44		df-dc 847	. 2 ⊢ (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
45	43, 44	sylibr 134	1 ⊢ (({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) ∈ Fin → DECID ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 720  DECID wdc 846   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {crab 2532   ∖ cdif 3217  ∅c0 3520  {csn 3709  1_oc1o 6684  Fincfn 7026

This proof depends on 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 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  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-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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1o 6691  df-er 6811  df-en 7027  df-fin 7029

This theorem is used by:  wexmiddifxy  17251