Theorem 0elsucexmid 4663

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 4617	. . . 4 ⊢ {𝑦 ∈ {∅} ∣ 𝜑} ∈ On
2		0elsucexmid.1	. . . 4 ⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥
3		suceq 4499	. . . . . 6 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑦 ∈ {∅} ∣ 𝜑})
4	3	eleq2d 2301	. . . . 5 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → (∅ ∈ suc 𝑥 ↔ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
5	4	rspcv 2906	. . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∅ ∈ suc 𝑥 → ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
6	1, 2, 5	mp2 16	. . 3 ⊢ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}
7		0ex 4216	. . . 4 ⊢ ∅ ∈ V
8	7	elsuc 4503	. . 3 ⊢ (∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}))
9	6, 8	mpbi 145	. 2 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑})
10	7	snid 3700	. . . . 5 ⊢ ∅ ∈ {∅}
11		biidd 172	. . . . . 6 ⊢ (𝑦 = ∅ → (𝜑 ↔ 𝜑))
12	11	elrab3 2963	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
13	10, 12	ax-mp 5	. . . 4 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
14	13	biimpi 120	. . 3 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} → 𝜑)
15		ordtriexmidlem2 4618	. . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
16	15	eqcoms 2234	. . 3 ⊢ (∅ = {𝑦 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
17	14, 16	orim12i 766	. 2 ⊢ ((∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
18	9, 17	ax-mp 5	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 105   ∨ wo 715   = wceq 1397   ∈ wcel 2202  ∀wral 2510  {crab 2514  ∅c0 3494  {csn 3669  Oncon0 4460  suc csuc 4462

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468

This theorem is referenced by: (None)