Theorem sosng 4620

Description: Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)

Assertion

Ref	Expression
sosng	⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Proof of Theorem sosng

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sopo 4243	. . 3 ⊢ (𝑅 Or {𝐴} → 𝑅 Po {𝐴})
2		posng 4619	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
3	1, 2	syl5ib 153	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Or {𝐴} → ¬ 𝐴𝑅𝐴))
4	2	biimpar 295	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Po {𝐴})
5		ax-in2 605	. . . . . . . . 9 ⊢ (¬ 𝐴𝑅𝐴 → (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
6	5	adantr 274	. . . . . . . 8 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
7		elsni 3550	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
8		elsni 3550	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
9	7, 8	breqan12d 3953	. . . . . . . . . 10 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐴))
10	9	imbi1d 230	. . . . . . . . 9 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
11	10	adantl 275	. . . . . . . 8 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
12	6, 11	mpbird 166	. . . . . . 7 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
13	12	ralrimivw 2509	. . . . . 6 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
14	13	ralrimivva 2517	. . . . 5 ⊢ (¬ 𝐴𝑅𝐴 → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
15	14	adantl 275	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
16		df-iso 4227	. . . 4 ⊢ (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
17	4, 15, 16	sylanbrc 414	. . 3 ⊢ (((Rel 𝑅 ∧ 𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Or {𝐴})
18	17	ex 114	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (¬ 𝐴𝑅𝐴 → 𝑅 Or {𝐴}))
19	3, 18	impbid 128	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∈ wcel 1481  ∀wral 2417  Vcvv 2689  {csn 3532   class class class wbr 3937   Po wpo 4224   Or wor 4225  Rel wrel 4552

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-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

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-v 2691  df-sbc 2914  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-po 4226  df-iso 4227

This theorem is referenced by: (None)