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Theorem supsnti 6409
Description: The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)
Hypotheses
Ref Expression
supsnti.ti ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
supsnti.b (𝜑𝐵𝐴)
Assertion
Ref Expression
supsnti (𝜑 → sup({𝐵}, 𝐴, 𝑅) = 𝐵)
Distinct variable groups:   𝑢,𝐴,𝑣   𝑢,𝐵,𝑣   𝑢,𝑅,𝑣   𝜑,𝑢,𝑣

Proof of Theorem supsnti
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 supsnti.ti . 2 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
2 supsnti.b . 2 (𝜑𝐵𝐴)
3 snidg 3428 . . 3 (𝐵𝐴𝐵 ∈ {𝐵})
42, 3syl 14 . 2 (𝜑𝐵 ∈ {𝐵})
5 eqid 2056 . . . . . 6 𝐵 = 𝐵
61ralrimivva 2418 . . . . . . 7 (𝜑 → ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
7 eqeq1 2062 . . . . . . . . . 10 (𝑢 = 𝐵 → (𝑢 = 𝑣𝐵 = 𝑣))
8 breq1 3795 . . . . . . . . . . . 12 (𝑢 = 𝐵 → (𝑢𝑅𝑣𝐵𝑅𝑣))
98notbid 602 . . . . . . . . . . 11 (𝑢 = 𝐵 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝐵𝑅𝑣))
10 breq2 3796 . . . . . . . . . . . 12 (𝑢 = 𝐵 → (𝑣𝑅𝑢𝑣𝑅𝐵))
1110notbid 602 . . . . . . . . . . 11 (𝑢 = 𝐵 → (¬ 𝑣𝑅𝑢 ↔ ¬ 𝑣𝑅𝐵))
129, 11anbi12d 450 . . . . . . . . . 10 (𝑢 = 𝐵 → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵)))
137, 12bibi12d 228 . . . . . . . . 9 (𝑢 = 𝐵 → ((𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ (𝐵 = 𝑣 ↔ (¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵))))
14 eqeq2 2065 . . . . . . . . . 10 (𝑣 = 𝐵 → (𝐵 = 𝑣𝐵 = 𝐵))
15 breq2 3796 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝐵𝑅𝑣𝐵𝑅𝐵))
1615notbid 602 . . . . . . . . . . 11 (𝑣 = 𝐵 → (¬ 𝐵𝑅𝑣 ↔ ¬ 𝐵𝑅𝐵))
17 breq1 3795 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑣𝑅𝐵𝐵𝑅𝐵))
1817notbid 602 . . . . . . . . . . 11 (𝑣 = 𝐵 → (¬ 𝑣𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
1916, 18anbi12d 450 . . . . . . . . . 10 (𝑣 = 𝐵 → ((¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵) ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵)))
2014, 19bibi12d 228 . . . . . . . . 9 (𝑣 = 𝐵 → ((𝐵 = 𝑣 ↔ (¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵)) ↔ (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))))
2113, 20rspc2v 2685 . . . . . . . 8 ((𝐵𝐴𝐵𝐴) → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) → (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))))
222, 2, 21syl2anc 397 . . . . . . 7 (𝜑 → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) → (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))))
236, 22mpd 13 . . . . . 6 (𝜑 → (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵)))
245, 23mpbii 140 . . . . 5 (𝜑 → (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))
2524simpld 109 . . . 4 (𝜑 → ¬ 𝐵𝑅𝐵)
2625adantr 265 . . 3 ((𝜑𝑥 ∈ {𝐵}) → ¬ 𝐵𝑅𝐵)
27 elsni 3421 . . . . . 6 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
2827breq2d 3804 . . . . 5 (𝑥 ∈ {𝐵} → (𝐵𝑅𝑥𝐵𝑅𝐵))
2928notbid 602 . . . 4 (𝑥 ∈ {𝐵} → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
3029adantl 266 . . 3 ((𝜑𝑥 ∈ {𝐵}) → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
3126, 30mpbird 160 . 2 ((𝜑𝑥 ∈ {𝐵}) → ¬ 𝐵𝑅𝑥)
321, 2, 4, 31supmaxti 6408 1 (𝜑 → sup({𝐵}, 𝐴, 𝑅) = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323  {csn 3403   class class class wbr 3792  supcsup 6388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-riota 5496  df-sup 6390
This theorem is referenced by: (None)
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