Theorem ssfiexmidt 7174

Description: If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.)

Assertion

Ref	Expression
ssfiexmidt	⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) → (𝜑 ∨ ¬ 𝜑))

Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ssfiexmidt

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		p0ex 4323	. . . 4 ⊢ {∅} ∈ V
2		eleq1 2301	. . . . . . 7 ⊢ (𝑥 = {∅} → (𝑥 ∈ Fin ↔ {∅} ∈ Fin))
3		sseq2 3272	. . . . . . 7 ⊢ (𝑥 = {∅} → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ {∅}))
4	2, 3	anbi12d 477	. . . . . 6 ⊢ (𝑥 = {∅} → ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) ↔ ({∅} ∈ Fin ∧ 𝑦 ⊆ {∅})))
5	4	imbi1d 231	. . . . 5 ⊢ (𝑥 = {∅} → (((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) ↔ (({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin)))
6	5	albidv 1877	. . . 4 ⊢ (𝑥 = {∅} → (∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) ↔ ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin)))
7	1, 6	spcv 2919	. . 3 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) → ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin))
8		0ex 4258	. . . . 5 ⊢ ∅ ∈ V
9		snfig 7097	. . . . 5 ⊢ (∅ ∈ V → {∅} ∈ Fin)
10	8, 9	ax-mp 5	. . . 4 ⊢ {∅} ∈ Fin
11		ssrab2 3333	. . . 4 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
12	10, 11	pm3.2i 272	. . 3 ⊢ ({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅})
13	1	rabex 4278	. . . 4 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
14		sseq1 3271	. . . . . 6 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
15	14	anbi2d 468	. . . . 5 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) ↔ ({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅})))
16		eleq1 2301	. . . . 5 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ∈ Fin ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin))
17	15, 16	imbi12d 234	. . . 4 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → ((({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin) ↔ (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin)))
18	13, 17	spcv 2919	. . 3 ⊢ (∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin) → (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin))
19	7, 12, 18	mpisyl 1496	. 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin)
20	19	ssfilemd 7173	1 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) → (𝜑 ∨ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  ∀wal 1400   = wceq 1402   ∈ wcel 2209  {crab 2532  Vcvv 2821   ⊆ wss 3220  ∅c0 3520  {csn 3708  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-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-iinf 4733

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-id 4436  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:  exmidssfi  7240