Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  oninfunirab Structured version   Visualization version   GIF version

Theorem oninfunirab 42289
Description: The infimum of a non-empty class of ordinals is the union of every ordinal less-than-or-equal to every element of that class. (Contributed by RP, 23-Jan-2025.)
Assertion
Ref Expression
oninfunirab ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑥𝑦})
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem oninfunirab
StepHypRef Expression
1 oninfint 42288 . 2 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = 𝐴)
2 onintunirab 42279 . 2 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑥𝑦})
31, 2eqtrd 2771 1 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑥𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wne 2939  wral 3060  {crab 3431  wss 3948  c0 4322   cuni 4908   cint 4950   E cep 5579  Oncon0 6364  infcinf 9440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-cnv 5684  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-riota 7368  df-sup 9441  df-inf 9442
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator