Theorem 0elsucexmid 4409

Description: If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)

Hypothesis

Ref	Expression
0elsucexmid.1	⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥

Assertion

Ref	Expression
0elsucexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem 0elsucexmid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtriexmidlem 4364	. . . 4 ⊢ {𝑦 ∈ {∅} ∣ 𝜑} ∈ On
2		0elsucexmid.1	. . . 4 ⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥
3		suceq 4253	. . . . . 6 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑦 ∈ {∅} ∣ 𝜑})
4	3	eleq2d 2164	. . . . 5 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → (∅ ∈ suc 𝑥 ↔ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
5	4	rspcv 2732	. . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∅ ∈ suc 𝑥 → ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
6	1, 2, 5	mp2 16	. . 3 ⊢ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}
7		0ex 3987	. . . 4 ⊢ ∅ ∈ V
8	7	elsuc 4257	. . 3 ⊢ (∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}))
9	6, 8	mpbi 144	. 2 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑})
10	7	snid 3495	. . . . 5 ⊢ ∅ ∈ {∅}
11		biidd 171	. . . . . 6 ⊢ (𝑦 = ∅ → (𝜑 ↔ 𝜑))
12	11	elrab3 2786	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
13	10, 12	ax-mp 7	. . . 4 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
14	13	biimpi 119	. . 3 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} → 𝜑)
15		ordtriexmidlem2 4365	. . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
16	15	eqcoms 2098	. . 3 ⊢ (∅ = {𝑦 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
17	14, 16	orim12i 714	. 2 ⊢ ((∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
18	9, 17	ax-mp 7	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 104   ∨ wo 667   = wceq 1296   ∈ wcel 1445  ∀wral 2370  {crab 2374  ∅c0 3302  {csn 3466  Oncon0 4214  suc csuc 4216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219  df-suc 4222

This theorem is referenced by: (None)