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Theorem elon2 5895
Description: An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
Assertion
Ref Expression
elon2 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))

Proof of Theorem elon2
StepHypRef Expression
1 elex 3352 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
2 elong 5892 . . 3 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2biadan2 677 . 2 (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴))
4 ancom 465 . 2 ((𝐴 ∈ V ∧ Ord 𝐴) ↔ (Ord 𝐴𝐴 ∈ V))
53, 4bitri 264 1 (𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2139  Vcvv 3340  Ord word 5883  Oncon0 5884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-uni 4589  df-tr 4905  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888
This theorem is referenced by:  sucelon  7182  tfrlem12  7654  tfrlem13  7655  gruina  9832  bdayimaon  32149
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