Theorem wexmiddifxy 17046

Description: Being able to subtract an arbitrary finite set from a finite set and get a finite set is equivalent to weak excluded middle. By adding additional conditions we can get a theorem which does not need weak excluded middle, at diffifi 7198. (Contributed by Jim Kingdon, 1-Aug-2026.)

Assertion

Ref	Expression
wexmiddifxy	⊢ (WEXMID ↔ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin))

Distinct variable group:   𝑥,𝑦

Proof of Theorem wexmiddifxy

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

Step	Hyp	Ref	Expression
1		wexmiddiffi 17044	. . 3 ⊢ (WEXMID ↔ ∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin))
2		pm3.41 331	. . . 4 ⊢ ((𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin))
3	2	2alimi 1509	. . 3 ⊢ (∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin))
4	1, 3	sylbi 121	. 2 ⊢ (WEXMID → ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin))
5		1oex 6695	. . . . . . . 8 ⊢ 1o ∈ V
6	5	snex 4322	. . . . . . 7 ⊢ {1o} ∈ V
7		eleq1 2301	. . . . . . . . . 10 ⊢ (𝑥 = {1o} → (𝑥 ∈ Fin ↔ {1o} ∈ Fin))
8	7	anbi1d 469	. . . . . . . . 9 ⊢ (𝑥 = {1o} → ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) ↔ ({1o} ∈ Fin ∧ 𝑦 ∈ Fin)))
9		difeq1 3340	. . . . . . . . . 10 ⊢ (𝑥 = {1o} → (𝑥 ∖ 𝑦) = ({1o} ∖ 𝑦))
10	9	eleq1d 2307	. . . . . . . . 9 ⊢ (𝑥 = {1o} → ((𝑥 ∖ 𝑦) ∈ Fin ↔ ({1o} ∖ 𝑦) ∈ Fin))
11	8, 10	imbi12d 234	. . . . . . . 8 ⊢ (𝑥 = {1o} → (((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin) ↔ (({1o} ∈ Fin ∧ 𝑦 ∈ Fin) → ({1o} ∖ 𝑦) ∈ Fin)))
12	11	albidv 1877	. . . . . . 7 ⊢ (𝑥 = {1o} → (∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin) ↔ ∀𝑦(({1o} ∈ Fin ∧ 𝑦 ∈ Fin) → ({1o} ∖ 𝑦) ∈ Fin)))
13	6, 12	spcv 2919	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin) → ∀𝑦(({1o} ∈ Fin ∧ 𝑦 ∈ Fin) → ({1o} ∖ 𝑦) ∈ Fin))
14		snfig 7103	. . . . . . . 8 ⊢ (1o ∈ V → {1o} ∈ Fin)
15	5, 14	ax-mp 5	. . . . . . 7 ⊢ {1o} ∈ Fin
16	5	rabex 4280	. . . . . . . 8 ⊢ {𝑧 ∈ 1o ∣ 𝑝 = 1o} ∈ V
17		snfig 7103	. . . . . . . 8 ⊢ ({𝑧 ∈ 1o ∣ 𝑝 = 1o} ∈ V → {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} ∈ Fin)
18	16, 17	ax-mp 5	. . . . . . 7 ⊢ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} ∈ Fin
19	15, 18	pm3.2i 272	. . . . . 6 ⊢ ({1o} ∈ Fin ∧ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} ∈ Fin)
20	16	snex 4322	. . . . . . 7 ⊢ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} ∈ V
21		eleq1 2301	. . . . . . . . 9 ⊢ (𝑦 = {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} → (𝑦 ∈ Fin ↔ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} ∈ Fin))
22	21	anbi2d 468	. . . . . . . 8 ⊢ (𝑦 = {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} → (({1o} ∈ Fin ∧ 𝑦 ∈ Fin) ↔ ({1o} ∈ Fin ∧ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} ∈ Fin)))
23		difeq2 3341	. . . . . . . . 9 ⊢ (𝑦 = {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} → ({1o} ∖ 𝑦) = ({1o} ∖ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}}))
24	23	eleq1d 2307	. . . . . . . 8 ⊢ (𝑦 = {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} → (({1o} ∖ 𝑦) ∈ Fin ↔ ({1o} ∖ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}}) ∈ Fin))
25	22, 24	imbi12d 234	. . . . . . 7 ⊢ (𝑦 = {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} → ((({1o} ∈ Fin ∧ 𝑦 ∈ Fin) → ({1o} ∖ 𝑦) ∈ Fin) ↔ (({1o} ∈ Fin ∧ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} ∈ Fin) → ({1o} ∖ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}}) ∈ Fin)))
26	20, 25	spcv 2919	. . . . . 6 ⊢ (∀𝑦(({1o} ∈ Fin ∧ 𝑦 ∈ Fin) → ({1o} ∖ 𝑦) ∈ Fin) → (({1o} ∈ Fin ∧ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}} ∈ Fin) → ({1o} ∖ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}}) ∈ Fin))
27	13, 19, 26	mpisyl 1496	. . . . 5 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin) → ({1o} ∖ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}}) ∈ Fin)
28		wexmiddifxylem 17045	. . . . 5 ⊢ (({1o} ∖ {{𝑧 ∈ 1o ∣ 𝑝 = 1o}}) ∈ Fin → DECID ¬ 𝑝 = 1o)
29		exmiddc 848	. . . . 5 ⊢ (DECID ¬ 𝑝 = 1o → (¬ 𝑝 = 1o ∨ ¬ ¬ 𝑝 = 1o))
30	27, 28, 29	3syl 17	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin) → (¬ 𝑝 = 1o ∨ ¬ ¬ 𝑝 = 1o))
31	30	ralrimivw 2624	. . 3 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin) → ∀𝑝 ∈ 𝒫 1o(¬ 𝑝 = 1o ∨ ¬ ¬ 𝑝 = 1o))
32		df-wexmid 17041	. . 3 ⊢ (WEXMID ↔ ∀𝑝 ∈ 𝒫 1o(¬ 𝑝 = 1o ∨ ¬ ¬ 𝑝 = 1o))
33	31, 32	sylibr 134	. 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin) → WEXMID)
34	4, 33	impbii 126	1 ⊢ (WEXMID ↔ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846  ∀wal 1400   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   ∖ cdif 3217  𝒫 cpw 3688  {csn 3709  1_oc1o 6680  Fincfn 7022  WEXMIDwwem 17040

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-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  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-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  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  df-wexmid 17041

This theorem is used by: (None)