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Theorem supsnti 6978
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 3610 . . 3 (𝐵𝐴𝐵 ∈ {𝐵})
42, 3syl 14 . 2 (𝜑𝐵 ∈ {𝐵})
5 eqid 2170 . . . . . 6 𝐵 = 𝐵
61ralrimivva 2552 . . . . . . 7 (𝜑 → ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
7 eqeq1 2177 . . . . . . . . . 10 (𝑢 = 𝐵 → (𝑢 = 𝑣𝐵 = 𝑣))
8 breq1 3990 . . . . . . . . . . . 12 (𝑢 = 𝐵 → (𝑢𝑅𝑣𝐵𝑅𝑣))
98notbid 662 . . . . . . . . . . 11 (𝑢 = 𝐵 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝐵𝑅𝑣))
10 breq2 3991 . . . . . . . . . . . 12 (𝑢 = 𝐵 → (𝑣𝑅𝑢𝑣𝑅𝐵))
1110notbid 662 . . . . . . . . . . 11 (𝑢 = 𝐵 → (¬ 𝑣𝑅𝑢 ↔ ¬ 𝑣𝑅𝐵))
129, 11anbi12d 470 . . . . . . . . . 10 (𝑢 = 𝐵 → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵)))
137, 12bibi12d 234 . . . . . . . . 9 (𝑢 = 𝐵 → ((𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ (𝐵 = 𝑣 ↔ (¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵))))
14 eqeq2 2180 . . . . . . . . . 10 (𝑣 = 𝐵 → (𝐵 = 𝑣𝐵 = 𝐵))
15 breq2 3991 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝐵𝑅𝑣𝐵𝑅𝐵))
1615notbid 662 . . . . . . . . . . 11 (𝑣 = 𝐵 → (¬ 𝐵𝑅𝑣 ↔ ¬ 𝐵𝑅𝐵))
17 breq1 3990 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑣𝑅𝐵𝐵𝑅𝐵))
1817notbid 662 . . . . . . . . . . 11 (𝑣 = 𝐵 → (¬ 𝑣𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
1916, 18anbi12d 470 . . . . . . . . . 10 (𝑣 = 𝐵 → ((¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵) ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵)))
2014, 19bibi12d 234 . . . . . . . . 9 (𝑣 = 𝐵 → ((𝐵 = 𝑣 ↔ (¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵)) ↔ (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))))
2113, 20rspc2v 2847 . . . . . . . 8 ((𝐵𝐴𝐵𝐴) → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) → (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))))
222, 2, 21syl2anc 409 . . . . . . 7 (𝜑 → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) → (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))))
236, 22mpd 13 . . . . . 6 (𝜑 → (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵)))
245, 23mpbii 147 . . . . 5 (𝜑 → (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))
2524simpld 111 . . . 4 (𝜑 → ¬ 𝐵𝑅𝐵)
2625adantr 274 . . 3 ((𝜑𝑥 ∈ {𝐵}) → ¬ 𝐵𝑅𝐵)
27 elsni 3599 . . . . . 6 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
2827breq2d 3999 . . . . 5 (𝑥 ∈ {𝐵} → (𝐵𝑅𝑥𝐵𝑅𝐵))
2928notbid 662 . . . 4 (𝑥 ∈ {𝐵} → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
3029adantl 275 . . 3 ((𝜑𝑥 ∈ {𝐵}) → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
3126, 30mpbird 166 . 2 ((𝜑𝑥 ∈ {𝐵}) → ¬ 𝐵𝑅𝑥)
321, 2, 4, 31supmaxti 6977 1 (𝜑 → sup({𝐵}, 𝐴, 𝑅) = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104   = wceq 1348  wcel 2141  wral 2448  {csn 3581   class class class wbr 3987  supcsup 6955
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-iota 5158  df-riota 5806  df-sup 6957
This theorem is referenced by:  infsnti  7003
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