Theorem ordtri2or2exmid 4572

Description: Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)

Hypothesis

Ref	Expression
ordtri2or2exmid.1	⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)

Assertion

Ref	Expression
ordtri2or2exmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ordtri2or2exmid

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtri2or2exmid.1	. . . 4 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)
2		ordtri2or2exmidlem 4527	. . . . 5 ⊢ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
3		suc0 4413	. . . . . 6 ⊢ suc ∅ = {∅}
4		0elon 4394	. . . . . . 7 ⊢ ∅ ∈ On
5	4	onsuci 4517	. . . . . 6 ⊢ suc ∅ ∈ On
6	3, 5	eqeltrri 2251	. . . . 5 ⊢ {∅} ∈ On
7		sseq1 3180	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑥 ⊆ 𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦))
8		sseq2 3181	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
9	7, 8	orbi12d 793	. . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ∨ 𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
10		sseq2 3181	. . . . . . 7 ⊢ (𝑦 = {∅} → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅}))
11		sseq1 3180	. . . . . . 7 ⊢ (𝑦 = {∅} → (𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
12	10, 11	orbi12d 793	. . . . . 6 ⊢ (𝑦 = {∅} → (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ∨ 𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
13	9, 12	rspc2va 2857	. . . . 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		elirr 4542	. . . . 5 ⊢ ¬ {∅} ∈ {∅}
17		simpl 109	. . . . . . 7 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅})
18		simpr 110	. . . . . . . 8 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → 𝜑)
19		p0ex 4190	. . . . . . . . . 10 ⊢ {∅} ∈ V
20	19	prid2 3701	. . . . . . . . 9 ⊢ {∅} ∈ {∅, {∅}}
21		biidd 172	. . . . . . . . . 10 ⊢ (𝑧 = {∅} → (𝜑 ↔ 𝜑))
22	21	elrab3 2896	. . . . . . . . 9 ⊢ ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
23	20, 22	ax-mp 5	. . . . . . . 8 ⊢ ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
24	18, 23	sylibr 134	. . . . . . 7 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
25	17, 24	sseldd 3158	. . . . . 6 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {∅})
26	25	ex 115	. . . . 5 ⊢ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → (𝜑 → {∅} ∈ {∅}))
27	16, 26	mtoi 664	. . . 4 ⊢ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → ¬ 𝜑)
28		snssg 3728	. . . . . 6 ⊢ (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
29	4, 28	ax-mp 5	. . . . 5 ⊢ (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
30		0ex 4132	. . . . . . . 8 ⊢ ∅ ∈ V
31	30	prid1 3700	. . . . . . 7 ⊢ ∅ ∈ {∅, {∅}}
32		biidd 172	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
33	32	elrab3 2896	. . . . . . 7 ⊢ (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
34	31, 33	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
35	34	biimpi 120	. . . . 5 ⊢ (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
36	29, 35	sylbir 135	. . . 4 ⊢ ({∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
37	27, 36	orim12i 759	. . 3 ⊢ (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑))
38	15, 37	ax-mp 5	. 2 ⊢ (¬ 𝜑 ∨ 𝜑)
39		orcom 728	. 2 ⊢ ((¬ 𝜑 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
40	38, 39	mpbi 145	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 708   = wceq 1353   ∈ wcel 2148  ∀wral 2455  {crab 2459   ⊆ wss 3131  ∅c0 3424  {csn 3594  {cpr 3595  Oncon0 4365  suc csuc 4367

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370  df-suc 4373

This theorem is referenced by:  onintexmid  4574