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Theorem elon 5893
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
Hypothesis
Ref Expression
elon.1 𝐴 ∈ V
Assertion
Ref Expression
elon (𝐴 ∈ On ↔ Ord 𝐴)

Proof of Theorem elon
StepHypRef Expression
1 elon.1 . 2 𝐴 ∈ V
2 elong 5892 . 2 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2ax-mp 5 1 (𝐴 ∈ On ↔ Ord 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  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:  tron  5907  0elon  5939  smogt  7634  dfrecs3  7639  rdglim2  7698  omeulem1  7833  isfinite2  8385  r0weon  9045  cflim3  9296  inar1  9809  ellimits  32344  dford3lem2  38114
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