Theorem omelon2 7804

Description: Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)

Assertion

Ref	Expression
omelon2	⊢ (ω ∈ V → ω ∈ On)

Proof of Theorem omelon2

Step	Hyp	Ref	Expression
1		omon 7803	. . . 4 ⊢ (ω ∈ On ∨ ω = On)
2	1	ori 861	. . 3 ⊢ (¬ ω ∈ On → ω = On)
3		onprc 7706	. . . 4 ⊢ ¬ On ∈ V
4		eleq1 2817	. . . 4 ⊢ (ω = On → (ω ∈ V ↔ On ∈ V))
5	3, 4	mtbiri 327	. . 3 ⊢ (ω = On → ¬ ω ∈ V)
6	2, 5	syl 17	. 2 ⊢ (¬ ω ∈ On → ¬ ω ∈ V)
7	6	con4i 114	1 ⊢ (ω ∈ V → ω ∈ On)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2110  Vcvv 3434  Oncon0 6302  ωcom 7791

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 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-tr 5197  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6305  df-on 6306  df-lim 6307  df-om 7792

This theorem is referenced by:  oaabs  8558  omelon  9531  fictb  10127  axdc3lem  10333