Theorem ordtri2orexmid 4647

Description: Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)

Hypothesis

Ref	Expression
ordtri2orexmid.1	⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑦 ⊆ 𝑥)

Assertion

Ref	Expression
ordtri2orexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ordtri2orexmid

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtri2orexmid.1	. . . 4 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑦 ⊆ 𝑥)
2		ordtriexmidlem 4643	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3		suc0 4534	. . . . . 6 ⊢ suc ∅ = {∅}
4		0elon 4515	. . . . . . 7 ⊢ ∅ ∈ On
5	4	onsuci 4640	. . . . . 6 ⊢ suc ∅ ∈ On
6	3, 5	eqeltrri 2308	. . . . 5 ⊢ {∅} ∈ On
7		eleq1 2297	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦))
8		sseq2 3264	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
9	7, 8	orbi12d 801	. . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ∨ 𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑})))
10		eleq2 2298	. . . . . . 7 ⊢ (𝑦 = {∅} → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅}))
11		sseq1 3263	. . . . . . 7 ⊢ (𝑦 = {∅} → (𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
12	10, 11	orbi12d 801	. . . . . 6 ⊢ (𝑦 = {∅} → (({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ∨ 𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑})))
13	9, 12	rspc2va 2937	. . . . 5 ⊢ ((({𝑧 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑦 ⊆ 𝑥)) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
14	2, 6, 13	mpanl12 436	. . . 4 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
15	1, 14	ax-mp 5	. . 3 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑})
16		elsni 3709	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
17		ordtriexmidlem2 4644	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
18	16, 17	syl 14	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
19		snssg 3830	. . . . . 6 ⊢ (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
20	4, 19	ax-mp 5	. . . . 5 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑})
21		0ex 4239	. . . . . . . 8 ⊢ ∅ ∈ V
22	21	snid 3722	. . . . . . 7 ⊢ ∅ ∈ {∅}
23		biidd 172	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
24	23	elrab3 2976	. . . . . . 7 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
25	22, 24	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
26	25	biimpi 120	. . . . 5 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
27	20, 26	sylbir 135	. . . 4 ⊢ ({∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
28	18, 27	orim12i 767	. . 3 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑))
29	15, 28	ax-mp 5	. 2 ⊢ (¬ 𝜑 ∨ 𝜑)
30		orcom 736	. 2 ⊢ ((¬ 𝜑 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
31	29, 30	mpbi 145	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2205  ∀wral 2522  {crab 2526   ⊆ wss 3213  ∅c0 3510  {csn 3691  Oncon0 4486  suc csuc 4488

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-uni 3917  df-tr 4211  df-iord 4489  df-on 4491  df-suc 4494

This theorem is referenced by: (None)