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Theorem uniordint 7741
Description: The union of a set of ordinals is equal to the intersection of its upper bounds. Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
Hypothesis
Ref Expression
uniordint.1 𝐴 ∈ V
Assertion
Ref Expression
uniordint (𝐴 ⊆ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem uniordint
StepHypRef Expression
1 uniordint.1 . . 3 𝐴 ∈ V
21ssonunii 7720 . 2 (𝐴 ⊆ On → 𝐴 ∈ On)
3 intmin 4934 . . 3 ( 𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = 𝐴)
4 unissb 4905 . . . . 5 ( 𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
54rabbii 3416 . . . 4 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
65inteqi 4916 . . 3 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
73, 6eqtr3di 2792 . 2 ( 𝐴 ∈ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
82, 7syl 17 1 (𝐴 ⊆ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wral 3065  {crab 3410  Vcvv 3448  wss 3915   cuni 4870   cint 4912  Oncon0 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by: (None)
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