Theorem ordtri2orexmid 4500

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 4496	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3		suc0 4389	. . . . . 6 ⊢ suc ∅ = {∅}
4		0elon 4370	. . . . . . 7 ⊢ ∅ ∈ On
5	4	onsuci 4493	. . . . . 6 ⊢ suc ∅ ∈ On
6	3, 5	eqeltrri 2240	. . . . 5 ⊢ {∅} ∈ On
7		eleq1 2229	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦))
8		sseq2 3166	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
9	7, 8	orbi12d 783	. . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ∨ 𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑})))
10		eleq2 2230	. . . . . . 7 ⊢ (𝑦 = {∅} → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅}))
11		sseq1 3165	. . . . . . 7 ⊢ (𝑦 = {∅} → (𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
12	10, 11	orbi12d 783	. . . . . 6 ⊢ (𝑦 = {∅} → (({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ∨ 𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑})))
13	9, 12	rspc2va 2844	. . . . 5 ⊢ ((({𝑧 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑦 ⊆ 𝑥)) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
14	2, 6, 13	mpanl12 433	. . . 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 3594	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
17		ordtriexmidlem2 4497	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
18	16, 17	syl 14	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
19		snssg 3709	. . . . . 6 ⊢ (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
20	4, 19	ax-mp 5	. . . . 5 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑})
21		0ex 4109	. . . . . . . 8 ⊢ ∅ ∈ V
22	21	snid 3607	. . . . . . 7 ⊢ ∅ ∈ {∅}
23		biidd 171	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
24	23	elrab3 2883	. . . . . . 7 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
25	22, 24	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
26	25	biimpi 119	. . . . 5 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
27	20, 26	sylbir 134	. . . 4 ⊢ ({∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
28	18, 27	orim12i 749	. . 3 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑))
29	15, 28	ax-mp 5	. 2 ⊢ (¬ 𝜑 ∨ 𝜑)
30		orcom 718	. 2 ⊢ ((¬ 𝜑 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
31	29, 30	mpbi 144	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 104   ∨ wo 698   = wceq 1343   ∈ wcel 2136  ∀wral 2444  {crab 2448   ⊆ wss 3116  ∅c0 3409  {csn 3576  Oncon0 4341  suc csuc 4343

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by: (None)