Theorem 0elsucexmid 4488

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 4443	. . . 4 ⊢ {𝑦 ∈ {∅} ∣ 𝜑} ∈ On
2		0elsucexmid.1	. . . 4 ⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥
3		suceq 4332	. . . . . 6 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑦 ∈ {∅} ∣ 𝜑})
4	3	eleq2d 2210	. . . . 5 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → (∅ ∈ suc 𝑥 ↔ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
5	4	rspcv 2789	. . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∅ ∈ suc 𝑥 → ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
6	1, 2, 5	mp2 16	. . 3 ⊢ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}
7		0ex 4063	. . . 4 ⊢ ∅ ∈ V
8	7	elsuc 4336	. . 3 ⊢ (∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}))
9	6, 8	mpbi 144	. 2 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑})
10	7	snid 3563	. . . . 5 ⊢ ∅ ∈ {∅}
11		biidd 171	. . . . . 6 ⊢ (𝑦 = ∅ → (𝜑 ↔ 𝜑))
12	11	elrab3 2845	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
13	10, 12	ax-mp 5	. . . 4 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
14	13	biimpi 119	. . 3 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} → 𝜑)
15		ordtriexmidlem2 4444	. . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
16	15	eqcoms 2143	. . 3 ⊢ (∅ = {𝑦 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
17	14, 16	orim12i 749	. 2 ⊢ ((∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
18	9, 17	ax-mp 5	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 104   ∨ wo 698   = wceq 1332   ∈ wcel 1481  ∀wral 2417  {crab 2421  ∅c0 3368  {csn 3532  Oncon0 4293  suc csuc 4295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-uni 3745  df-tr 4035  df-iord 4296  df-on 4298  df-suc 4301

This theorem is referenced by: (None)