Theorem wexmiddiffi 17210

Description: Being able to subtract an arbitrary 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, 29-Jul-2026.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem wexmiddiffi

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

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 ⊢ ((WEXMID ∧ 𝑥 ∈ Fin) → 𝑥 ∈ Fin)
2		difssd 3356	. . . . 5 ⊢ ((WEXMID ∧ 𝑥 ∈ Fin) → (𝑥 ∖ 𝑦) ⊆ 𝑥)
3		animorrl 838	. . . . . . . . 9 ⊢ (((WEXMID ∧ 𝑥 ∈ Fin) ∧ 𝑧 ∈ 𝑥) → (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥))
4		df-dc 847	. . . . . . . . 9 ⊢ (DECID 𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑥 ∨ ¬ 𝑧 ∈ 𝑥))
5	3, 4	sylibr 134	. . . . . . . 8 ⊢ (((WEXMID ∧ 𝑥 ∈ Fin) ∧ 𝑧 ∈ 𝑥) → DECID 𝑧 ∈ 𝑥)
6		wexmiddc 17208	. . . . . . . . 9 ⊢ (WEXMID → DECID ¬ 𝑧 ∈ 𝑦)
7	6	ad2antrr 492	. . . . . . . 8 ⊢ (((WEXMID ∧ 𝑥 ∈ Fin) ∧ 𝑧 ∈ 𝑥) → DECID ¬ 𝑧 ∈ 𝑦)
8	5, 7	dcand 945	. . . . . . 7 ⊢ (((WEXMID ∧ 𝑥 ∈ Fin) ∧ 𝑧 ∈ 𝑥) → DECID (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))
9		eldif 3229	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑥 ∖ 𝑦) ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))
10	9	dcbii 852	. . . . . . 7 ⊢ (DECID 𝑧 ∈ (𝑥 ∖ 𝑦) ↔ DECID (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))
11	8, 10	sylibr 134	. . . . . 6 ⊢ (((WEXMID ∧ 𝑥 ∈ Fin) ∧ 𝑧 ∈ 𝑥) → DECID 𝑧 ∈ (𝑥 ∖ 𝑦))
12	11	ralrimiva 2623	. . . . 5 ⊢ ((WEXMID ∧ 𝑥 ∈ Fin) → ∀𝑧 ∈ 𝑥 DECID 𝑧 ∈ (𝑥 ∖ 𝑦))
13		ssfidc 7245	. . . . 5 ⊢ ((𝑥 ∈ Fin ∧ (𝑥 ∖ 𝑦) ⊆ 𝑥 ∧ ∀𝑧 ∈ 𝑥 DECID 𝑧 ∈ (𝑥 ∖ 𝑦)) → (𝑥 ∖ 𝑦) ∈ Fin)
14	1, 2, 12, 13	syl3anc 1278	. . . 4 ⊢ ((WEXMID ∧ 𝑥 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin)
15	14	ex 115	. . 3 ⊢ (WEXMID → (𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin))
16	15	alrimivv 1928	. 2 ⊢ (WEXMID → ∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin))
17		wexmiddiffilem 17209	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → (¬ 𝑝 = 1o ∨ ¬ ¬ 𝑝 = 1o))
18	17	ralrimivw 2624	. . 3 ⊢ (∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → ∀𝑝 ∈ 𝒫 1o(¬ 𝑝 = 1o ∨ ¬ ¬ 𝑝 = 1o))
19		df-wexmid 17207	. . 3 ⊢ (WEXMID ↔ ∀𝑝 ∈ 𝒫 1o(¬ 𝑝 = 1o ∨ ¬ ¬ 𝑝 = 1o))
20	18, 19	sylibr 134	. 2 ⊢ (∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → WEXMID)
21	16, 20	impbii 126	1 ⊢ (WEXMID ↔ ∀𝑥∀𝑦(𝑥 ∈ 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   ∖ cdif 3217   ⊆ wss 3220  𝒫 cpw 3688  1_oc1o 6680  Fincfn 7022  WEXMIDwwem 17206

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 17207

This theorem is used by:  wexmiddifxy  17212