Theorem ontr2exmid 4539

Description: An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)

Hypothesis

Ref	Expression
ontr2exmid.1	⊢ ∀𝑥 ∈ On ∀𝑦∀𝑧 ∈ On ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)

Assertion

Ref	Expression
ontr2exmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑥,𝑦,𝑧

Proof of Theorem ontr2exmid

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3255	. . . . 5 ⊢ {𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2		p0ex 4203	. . . . . 6 ⊢ {∅} ∈ V
3	2	prid2 3714	. . . . 5 ⊢ {∅} ∈ {∅, {∅}}
4		2ordpr 4538	. . . . . . 7 ⊢ Ord {∅, {∅}}
5		pp0ex 4204	. . . . . . . 8 ⊢ {∅, {∅}} ∈ V
6	5	elon 4389	. . . . . . 7 ⊢ ({∅, {∅}} ∈ On ↔ Ord {∅, {∅}})
7	4, 6	mpbir 146	. . . . . 6 ⊢ {∅, {∅}} ∈ On
8		ordtriexmidlem 4533	. . . . . . . 8 ⊢ {𝑤 ∈ {∅} ∣ 𝜑} ∈ On
9		ontr2exmid.1	. . . . . . . 8 ⊢ ∀𝑥 ∈ On ∀𝑦∀𝑧 ∈ On ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)
10		sseq1 3193	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ 𝑦 ↔ {𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦))
11	10	anbi1d 465	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧)))
12		eleq1 2252	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ 𝑧 ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧))
13	11, 12	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) ↔ (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
14	13	ralbidv 2490	. . . . . . . . . 10 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (∀𝑧 ∈ On ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) ↔ ∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
15	14	albidv 1835	. . . . . . . . 9 ⊢ (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (∀𝑦∀𝑧 ∈ On ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) ↔ ∀𝑦∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
16	15	rspcv 2852	. . . . . . . 8 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∀𝑦∀𝑧 ∈ On ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) → ∀𝑦∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
17	8, 9, 16	mp2 16	. . . . . . 7 ⊢ ∀𝑦∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)
18		sseq2 3194	. . . . . . . . . . 11 ⊢ (𝑦 = {∅} → ({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ↔ {𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
19		eleq1 2252	. . . . . . . . . . 11 ⊢ (𝑦 = {∅} → (𝑦 ∈ 𝑧 ↔ {∅} ∈ 𝑧))
20	18, 19	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑦 = {∅} → (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧)))
21	20	imbi1d 231	. . . . . . . . 9 ⊢ (𝑦 = {∅} → ((({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) ↔ (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
22	21	ralbidv 2490	. . . . . . . 8 ⊢ (𝑦 = {∅} → (∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) ↔ ∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
23	2, 22	spcv 2846	. . . . . . 7 ⊢ (∀𝑦∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) → ∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧))
24	17, 23	ax-mp 5	. . . . . 6 ⊢ ∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)
25		eleq2 2253	. . . . . . . . 9 ⊢ (𝑧 = {∅, {∅}} → ({∅} ∈ 𝑧 ↔ {∅} ∈ {∅, {∅}}))
26	25	anbi2d 464	. . . . . . . 8 ⊢ (𝑧 = {∅, {∅}} → (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ {∅, {∅}})))
27		eleq2 2253	. . . . . . . 8 ⊢ (𝑧 = {∅, {∅}} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧 ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}}))
28	26, 27	imbi12d 234	. . . . . . 7 ⊢ (𝑧 = {∅, {∅}} → ((({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) ↔ (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ {∅, {∅}}) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}})))
29	28	rspcv 2852	. . . . . 6 ⊢ ({∅, {∅}} ∈ On → (∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) → (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ {∅, {∅}}) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}})))
30	7, 24, 29	mp2 16	. . . . 5 ⊢ (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ {∅, {∅}}) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}})
31	1, 3, 30	mp2an 426	. . . 4 ⊢ {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}}
32		elpri 3630	. . . 4 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}} → ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ ∨ {𝑤 ∈ {∅} ∣ 𝜑} = {∅}))
33	31, 32	ax-mp 5	. . 3 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ ∨ {𝑤 ∈ {∅} ∣ 𝜑} = {∅})
34		ordtriexmidlem2 4534	. . . 4 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
35		0ex 4145	. . . . 5 ⊢ ∅ ∈ V
36		biidd 172	. . . . 5 ⊢ (𝑤 = ∅ → (𝜑 ↔ 𝜑))
37	35, 36	rabsnt 3682	. . . 4 ⊢ ({𝑤 ∈ {∅} ∣ 𝜑} = {∅} → 𝜑)
38	34, 37	orim12i 760	. . 3 ⊢ (({𝑤 ∈ {∅} ∣ 𝜑} = ∅ ∨ {𝑤 ∈ {∅} ∣ 𝜑} = {∅}) → (¬ 𝜑 ∨ 𝜑))
39	33, 38	ax-mp 5	. 2 ⊢ (¬ 𝜑 ∨ 𝜑)
40		orcom 729	. 2 ⊢ ((¬ 𝜑 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
41	39, 40	mpbi 145	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  ∀wal 1362   = wceq 1364   ∈ wcel 2160  ∀wral 2468  {crab 2472   ⊆ wss 3144  ∅c0 3437  {csn 3607  {cpr 3608  Ord word 4377  Oncon0 4378

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 4189

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-pr 3614  df-uni 3825  df-tr 4117  df-iord 4381  df-on 4383  df-suc 4386

This theorem is referenced by: (None)