Theorem sosng 4696

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 4310	. . 3 ⊢ (𝑅 Or {𝐴} → 𝑅 Po {𝐴})
2		posng 4695	. . 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 615	. . . . . . . . 9 ⊢ (¬ 𝐴𝑅𝐴 → (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
6	5	adantr 276	. . . . . . . 8 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
7		elsni 3609	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
8		elsni 3609	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
9	7, 8	breqan12d 4016	. . . . . . . . . 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 2551	. . . . . 6 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
14	13	ralrimivva 2559	. . . . 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 4294	. . . 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 708   ∈ wcel 2148  ∀wral 2455  Vcvv 2737  {csn 3591   class class class wbr 4000   Po wpo 4291   Or wor 4292  Rel wrel 4628

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-ext 2159

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-ral 2460  df-v 2739  df-sbc 2963  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-po 4293  df-iso 4294

This theorem is referenced by: (None)