Theorem exmidssfi 7240

Description: Excluded middle is equivalent to any subset of a finite set being finite. Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 20-Mar-2026.)

Assertion

Ref	Expression
exmidssfi	⊢ (EXMID ↔ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin))

Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidssfi

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

Step	Hyp	Ref	Expression
1		simprl 535	. . . . 5 ⊢ ((EXMID ∧ (𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ Fin)
2		simprr 537	. . . . 5 ⊢ ((EXMID ∧ (𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
3		exmidexmid 4331	. . . . . . 7 ⊢ (EXMID → DECID 𝑤 ∈ 𝑦)
4	3	adantr 276	. . . . . 6 ⊢ ((EXMID ∧ (𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥)) → DECID 𝑤 ∈ 𝑦)
5	4	ralrimivw 2624	. . . . 5 ⊢ ((EXMID ∧ (𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥)) → ∀𝑤 ∈ 𝑥 DECID 𝑤 ∈ 𝑦)
6		ssfidc 7239	. . . . 5 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 DECID 𝑤 ∈ 𝑦) → 𝑦 ∈ Fin)
7	1, 2, 5, 6	syl3anc 1278	. . . 4 ⊢ ((EXMID ∧ (𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ Fin)
8	7	ex 115	. . 3 ⊢ (EXMID → ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin))
9	8	alrimivv 1928	. 2 ⊢ (EXMID → ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin))
10		ssfiexmidt 7174	. . . . 5 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) → (𝑧 = {∅} ∨ ¬ 𝑧 = {∅}))
11		df-dc 847	. . . . 5 ⊢ (DECID 𝑧 = {∅} ↔ (𝑧 = {∅} ∨ ¬ 𝑧 = {∅}))
12	10, 11	sylibr 134	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) → DECID 𝑧 = {∅})
13	12	adantr 276	. . 3 ⊢ ((∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) ∧ 𝑧 ⊆ {∅}) → DECID 𝑧 = {∅})
14	13	exmid1dc 4335	. 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) → EXMID)
15	9, 14	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846  ∀wal 1400   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  ∅c0 3520  {csn 3708  EXMIDwem 4329  Fincfn 7016

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-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-exmid 4330  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1o 6681  df-er 6801  df-en 7017  df-fin 7019

This theorem is referenced by: (None)