Theorem ssfiexmid 7058

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.)

Hypothesis

Ref	Expression
ssfiexmid.1	⊢ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin)

Assertion

Ref	Expression
ssfiexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ssfiexmid

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4214	. . . 4 ⊢ ∅ ∈ V
2		snfig 6984	. . . 4 ⊢ (∅ ∈ V → {∅} ∈ Fin)
3	1, 2	ax-mp 5	. . 3 ⊢ {∅} ∈ Fin
4		ssrab2 3310	. . 3 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
5		ssfiexmid.1	. . . . 5 ⊢ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin)
6		p0ex 4276	. . . . . 6 ⊢ {∅} ∈ V
7		eleq1 2292	. . . . . . . . 9 ⊢ (𝑥 = {∅} → (𝑥 ∈ Fin ↔ {∅} ∈ Fin))
8		sseq2 3249	. . . . . . . . 9 ⊢ (𝑥 = {∅} → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ {∅}))
9	7, 8	anbi12d 473	. . . . . . . 8 ⊢ (𝑥 = {∅} → ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) ↔ ({∅} ∈ Fin ∧ 𝑦 ⊆ {∅})))
10	9	imbi1d 231	. . . . . . 7 ⊢ (𝑥 = {∅} → (((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) ↔ (({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin)))
11	10	albidv 1870	. . . . . 6 ⊢ (𝑥 = {∅} → (∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) ↔ ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin)))
12	6, 11	spcv 2898	. . . . 5 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) → ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin))
13	5, 12	ax-mp 5	. . . 4 ⊢ ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin)
14	6	rabex 4232	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
15		sseq1 3248	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
16	15	anbi2d 464	. . . . . 6 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) ↔ ({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅})))
17		eleq1 2292	. . . . . 6 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ∈ Fin ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin))
18	16, 17	imbi12d 234	. . . . 5 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → ((({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin) ↔ (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin)))
19	14, 18	spcv 2898	. . . 4 ⊢ (∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin) → (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin))
20	13, 19	ax-mp 5	. . 3 ⊢ (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin)
21	3, 4, 20	mp2an 426	. 2 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin
22	21	ssfilem 7057	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 713  ∀wal 1393   = wceq 1395   ∈ wcel 2200  {crab 2512  Vcvv 2800   ⊆ wss 3198  ∅c0 3492  {csn 3667  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-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  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-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-id 4388  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:  infiexmid  7059