Theorem exmidssfi 7125

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 529	. . . . 5 ⊢ ((EXMID ∧ (𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ Fin)
2		simprr 531	. . . . 5 ⊢ ((EXMID ∧ (𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
3		exmidexmid 4284	. . . . . . 7 ⊢ (EXMID → DECID 𝑤 ∈ 𝑦)
4	3	adantr 276	. . . . . 6 ⊢ ((EXMID ∧ (𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥)) → DECID 𝑤 ∈ 𝑦)
5	4	ralrimivw 2604	. . . . 5 ⊢ ((EXMID ∧ (𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥)) → ∀𝑤 ∈ 𝑥 DECID 𝑤 ∈ 𝑦)
6		ssfidc 7124	. . . . 5 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 DECID 𝑤 ∈ 𝑦) → 𝑦 ∈ Fin)
7	1, 2, 5, 6	syl3anc 1271	. . . 4 ⊢ ((EXMID ∧ (𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ Fin)
8	7	ex 115	. . 3 ⊢ (EXMID → ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin))
9	8	alrimivv 1921	. 2 ⊢ (EXMID → ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin))
10		ssfiexmidt 7060	. . . . 5 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) → (𝑧 = {∅} ∨ ¬ 𝑧 = {∅}))
11		df-dc 840	. . . . 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 4288	. 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 713  DECID wdc 839  ∀wal 1393   = wceq 1395   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3198  ∅c0 3492  {csn 3667  EXMIDwem 4282  Fincfn 6904

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-exmid 4283  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1o 6577  df-er 6697  df-en 6905  df-fin 6907

This theorem is referenced by: (None)