Theorem sosng 4717

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 4331	. . 3 ⊢ (𝑅 Or {𝐴} → 𝑅 Po {𝐴})
2		posng 4716	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
3	1, 2	imbitrid 154	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Or {𝐴} → ¬ 𝐴𝑅𝐴))
4	2	biimpar 297	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Po {𝐴})
5		ax-in2 616	. . . . . . . . 9 ⊢ (¬ 𝐴𝑅𝐴 → (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
6	5	adantr 276	. . . . . . . 8 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
7		elsni 3625	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
8		elsni 3625	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
9	7, 8	breqan12d 4034	. . . . . . . . . 10 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐴))
10	9	imbi1d 231	. . . . . . . . 9 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
11	10	adantl 277	. . . . . . . 8 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
12	6, 11	mpbird 167	. . . . . . 7 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
13	12	ralrimivw 2564	. . . . . 6 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
14	13	ralrimivva 2572	. . . . 5 ⊢ (¬ 𝐴𝑅𝐴 → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
15	14	adantl 277	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
16		df-iso 4315	. . . 4 ⊢ (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
17	4, 15, 16	sylanbrc 417	. . 3 ⊢ (((Rel 𝑅 ∧ 𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Or {𝐴})
18	17	ex 115	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (¬ 𝐴𝑅𝐴 → 𝑅 Or {𝐴}))
19	3, 18	impbid 129	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∈ wcel 2160  ∀wral 2468  Vcvv 2752  {csn 3607   class class class wbr 4018   Po wpo 4312   Or wor 4313  Rel wrel 4649

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-ext 2171

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-v 2754  df-sbc 2978  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-po 4314  df-iso 4315

This theorem is referenced by: (None)