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Theorem onsupuni 42433
Description: The supremum of a set of ordinals is the union of that set. Lemma 2.10 of [Schloeder] p. 5. (Contributed by RP, 19-Jan-2025.)
Assertion
Ref Expression
onsupuni ((𝐴 ⊆ On ∧ 𝐴𝑉) → sup(𝐴, On, E ) = 𝐴)

Proof of Theorem onsupuni
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssonuni 7760 . . 3 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
21impcom 407 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → 𝐴 ∈ On)
3 elssuni 4931 . . . 4 (𝑦𝐴𝑦 𝐴)
43rgen 3055 . . 3 𝑦𝐴 𝑦 𝐴
5 simpl 482 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴𝑉) → 𝐴 ⊆ On)
65sselda 3974 . . . . . 6 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ 𝑦𝐴) → 𝑦 ∈ On)
72adantr 480 . . . . . 6 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ 𝑦𝐴) → 𝐴 ∈ On)
8 ontri1 6388 . . . . . 6 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 𝐴 ↔ ¬ 𝐴𝑦))
96, 7, 8syl2anc 583 . . . . 5 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ 𝑦𝐴) → (𝑦 𝐴 ↔ ¬ 𝐴𝑦))
10 epel 5573 . . . . . 6 ( 𝐴 E 𝑦 𝐴𝑦)
1110notbii 320 . . . . 5 𝐴 E 𝑦 ↔ ¬ 𝐴𝑦)
129, 11bitr4di 289 . . . 4 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ 𝑦𝐴) → (𝑦 𝐴 ↔ ¬ 𝐴 E 𝑦))
1312ralbidva 3167 . . 3 ((𝐴 ⊆ On ∧ 𝐴𝑉) → (∀𝑦𝐴 𝑦 𝐴 ↔ ∀𝑦𝐴 ¬ 𝐴 E 𝑦))
144, 13mpbii 232 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → ∀𝑦𝐴 ¬ 𝐴 E 𝑦)
152adantr 480 . . . . . 6 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
16 epelg 5571 . . . . . 6 ( 𝐴 ∈ On → (𝑦 E 𝐴𝑦 𝐴))
1715, 16syl 17 . . . . 5 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ 𝑦 ∈ On) → (𝑦 E 𝐴𝑦 𝐴))
1817biimpd 228 . . . 4 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ 𝑦 ∈ On) → (𝑦 E 𝐴𝑦 𝐴))
19 eluni2 4903 . . . . 5 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
20 epel 5573 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
2120rexbii 3086 . . . . 5 (∃𝑥𝐴 𝑦 E 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
2219, 21bitr4i 278 . . . 4 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦 E 𝑥)
2318, 22imbitrdi 250 . . 3 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ 𝑦 ∈ On) → (𝑦 E 𝐴 → ∃𝑥𝐴 𝑦 E 𝑥))
2423ralrimiva 3138 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → ∀𝑦 ∈ On (𝑦 E 𝐴 → ∃𝑥𝐴 𝑦 E 𝑥))
25 epweon 7755 . . . 4 E We On
26 weso 5657 . . . 4 ( E We On → E Or On)
2725, 26mp1i 13 . . 3 ((𝐴 ⊆ On ∧ 𝐴𝑉) → E Or On)
2827eqsup 9446 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → (( 𝐴 ∈ On ∧ ∀𝑦𝐴 ¬ 𝐴 E 𝑦 ∧ ∀𝑦 ∈ On (𝑦 E 𝐴 → ∃𝑥𝐴 𝑦 E 𝑥)) → sup(𝐴, On, E ) = 𝐴))
292, 14, 24, 28mp3and 1460 1 ((𝐴 ⊆ On ∧ 𝐴𝑉) → sup(𝐴, On, E ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3053  wrex 3062  wss 3940   cuni 4899   class class class wbr 5138   E cep 5569   Or wor 5577   We wwe 5620  Oncon0 6354  supcsup 9430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-tr 5256  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6357  df-on 6358  df-iota 6485  df-riota 7357  df-sup 9432
This theorem is referenced by:  onsupuni2  42434  onsupintrab  42435  limexissup  42486  limexissupab  42488
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