Theorem 0elsucexmid 4582

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 4536	. . . 4 ⊢ {𝑦 ∈ {∅} ∣ 𝜑} ∈ On
2		0elsucexmid.1	. . . 4 ⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥
3		suceq 4420	. . . . . 6 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑦 ∈ {∅} ∣ 𝜑})
4	3	eleq2d 2259	. . . . 5 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → (∅ ∈ suc 𝑥 ↔ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
5	4	rspcv 2852	. . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∅ ∈ suc 𝑥 → ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
6	1, 2, 5	mp2 16	. . 3 ⊢ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}
7		0ex 4145	. . . 4 ⊢ ∅ ∈ V
8	7	elsuc 4424	. . 3 ⊢ (∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}))
9	6, 8	mpbi 145	. 2 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑})
10	7	snid 3638	. . . . 5 ⊢ ∅ ∈ {∅}
11		biidd 172	. . . . . 6 ⊢ (𝑦 = ∅ → (𝜑 ↔ 𝜑))
12	11	elrab3 2909	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
13	10, 12	ax-mp 5	. . . 4 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
14	13	biimpi 120	. . 3 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} → 𝜑)
15		ordtriexmidlem2 4537	. . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
16	15	eqcoms 2192	. . 3 ⊢ (∅ = {𝑦 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
17	14, 16	orim12i 760	. 2 ⊢ ((∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
18	9, 17	ax-mp 5	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2160  ∀wral 2468  {crab 2472  ∅c0 3437  {csn 3607  Oncon0 4381  suc csuc 4383

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-uni 3825  df-tr 4117  df-iord 4384  df-on 4386  df-suc 4389

This theorem is referenced by: (None)