Theorem wexmiddiffilem 17043

Description: Lemma for wexmiddiffi 17044. The reverse direction, using different notation. (Contributed by Jim Kingdon, 29-Jul-2026.)

Assertion

Ref	Expression
wexmiddiffilem	⊢ (∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))

Distinct variable groups:   𝜑,𝑦   𝑥,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem wexmiddiffilem

Dummy variables 𝑗 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		p0ex 4325	. . . . 5 ⊢ {∅} ∈ V
2		eleq1 2301	. . . . . . 7 ⊢ (𝑥 = {∅} → (𝑥 ∈ Fin ↔ {∅} ∈ Fin))
3		difeq1 3340	. . . . . . . 8 ⊢ (𝑥 = {∅} → (𝑥 ∖ 𝑦) = ({∅} ∖ 𝑦))
4	3	eleq1d 2307	. . . . . . 7 ⊢ (𝑥 = {∅} → ((𝑥 ∖ 𝑦) ∈ Fin ↔ ({∅} ∖ 𝑦) ∈ Fin))
5	2, 4	imbi12d 234	. . . . . 6 ⊢ (𝑥 = {∅} → ((𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) ↔ ({∅} ∈ Fin → ({∅} ∖ 𝑦) ∈ Fin)))
6	5	albidv 1877	. . . . 5 ⊢ (𝑥 = {∅} → (∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) ↔ ∀𝑦({∅} ∈ Fin → ({∅} ∖ 𝑦) ∈ Fin)))
7	1, 6	spcv 2919	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → ∀𝑦({∅} ∈ Fin → ({∅} ∖ 𝑦) ∈ Fin))
8		0ex 4260	. . . . 5 ⊢ ∅ ∈ V
9		snfig 7103	. . . . 5 ⊢ (∅ ∈ V → {∅} ∈ Fin)
10	8, 9	ax-mp 5	. . . 4 ⊢ {∅} ∈ Fin
11	1	rabex 4280	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
12		difeq2 3341	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → ({∅} ∖ 𝑦) = ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
13	12	eleq1d 2307	. . . . . 6 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (({∅} ∖ 𝑦) ∈ Fin ↔ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) ∈ Fin))
14	13	imbi2d 230	. . . . 5 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (({∅} ∈ Fin → ({∅} ∖ 𝑦) ∈ Fin) ↔ ({∅} ∈ Fin → ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) ∈ Fin)))
15	11, 14	spcv 2919	. . . 4 ⊢ (∀𝑦({∅} ∈ Fin → ({∅} ∖ 𝑦) ∈ Fin) → ({∅} ∈ Fin → ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) ∈ Fin))
16	7, 10, 15	mpisyl 1496	. . 3 ⊢ (∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) ∈ Fin)
17		fin0or 7190	. . 3 ⊢ (({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) ∈ Fin → (({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) = ∅ ∨ ∃𝑗 𝑗 ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})))
18		notm0 3542	. . . . 5 ⊢ (¬ ∃𝑗 𝑗 ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) = ∅)
19	8	snm 3833	. . . . . . 7 ⊢ ∃𝑗 𝑗 ∈ {∅}
20	8	snm 3833	. . . . . . . . . . . . 13 ⊢ ∃𝑤 𝑤 ∈ {∅}
21		r19.3rmv 3618	. . . . . . . . . . . . 13 ⊢ (∃𝑤 𝑤 ∈ {∅} → (¬ 𝜑 ↔ ∀𝑧 ∈ {∅} ¬ 𝜑))
22	20, 21	ax-mp 5	. . . . . . . . . . . 12 ⊢ (¬ 𝜑 ↔ ∀𝑧 ∈ {∅} ¬ 𝜑)
23		rabeq0 3552	. . . . . . . . . . . 12 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ↔ ∀𝑧 ∈ {∅} ¬ 𝜑)
24	22, 23	sylbb2 138	. . . . . . . . . . 11 ⊢ (¬ 𝜑 → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
25	24	difeq2d 3347	. . . . . . . . . 10 ⊢ (¬ 𝜑 → ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) = ({∅} ∖ ∅))
26		dif0 3596	. . . . . . . . . 10 ⊢ ({∅} ∖ ∅) = {∅}
27	25, 26	eqtrdi 2287	. . . . . . . . 9 ⊢ (¬ 𝜑 → ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) = {∅})
28	27	eleq2d 2308	. . . . . . . 8 ⊢ (¬ 𝜑 → (𝑗 ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ 𝑗 ∈ {∅}))
29	28	exbidv 1878	. . . . . . 7 ⊢ (¬ 𝜑 → (∃𝑗 𝑗 ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ∃𝑗 𝑗 ∈ {∅}))
30	19, 29	mpbiri 168	. . . . . 6 ⊢ (¬ 𝜑 → ∃𝑗 𝑗 ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
31	30	con3i 641	. . . . 5 ⊢ (¬ ∃𝑗 𝑗 ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ ¬ 𝜑)
32	18, 31	sylbir 135	. . . 4 ⊢ (({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) = ∅ → ¬ ¬ 𝜑)
33		eldifi 3351	. . . . . 6 ⊢ (𝑗 ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → 𝑗 ∈ {∅})
34		eldifn 3352	. . . . . . 7 ⊢ (𝑗 ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ 𝑗 ∈ {𝑧 ∈ {∅} ∣ 𝜑})
35		biidd 172	. . . . . . . 8 ⊢ (𝑧 = 𝑗 → (𝜑 ↔ 𝜑))
36	35	elrab 2982	. . . . . . 7 ⊢ (𝑗 ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ (𝑗 ∈ {∅} ∧ 𝜑))
37	34, 36	sylnib 687	. . . . . 6 ⊢ (𝑗 ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ (𝑗 ∈ {∅} ∧ 𝜑))
38	33, 37	mpnanrd 704	. . . . 5 ⊢ (𝑗 ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ 𝜑)
39	38	exlimiv 1651	. . . 4 ⊢ (∃𝑗 𝑗 ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ 𝜑)
40	32, 39	orim12i 771	. . 3 ⊢ ((({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) = ∅ ∨ ∃𝑗 𝑗 ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) → (¬ ¬ 𝜑 ∨ ¬ 𝜑))
41	16, 17, 40	3syl 17	. 2 ⊢ (∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → (¬ ¬ 𝜑 ∨ ¬ 𝜑))
42	41	orcomd 741	1 ⊢ (∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  ∀wal 1400   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   ∖ cdif 3217  ∅c0 3520  {csn 3709  Fincfn 7022

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 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-iinf 4735

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1o 6687  df-er 6807  df-en 7023  df-fin 7025

This theorem is used by:  wexmiddiffi  17044