Theorem xoromon 35194

Description: ω is either an ordinal set or the proper class of all ordinal sets, but not both. This is a stronger version of omon 7818. (Contributed by BTernaryTau, 25-Jan-2026.)

Assertion

Ref	Expression
xoromon	⊢ (ω ∈ On ⊻ ω = On)

Proof of Theorem xoromon

Step	Hyp	Ref	Expression
1		omon 7818	. 2 ⊢ (ω ∈ On ∨ ω = On)
2		onprc 7721	. . . . . 6 ⊢ ¬ On ∈ V
3		prcnel 3464	. . . . . 6 ⊢ (¬ On ∈ V → ¬ On ∈ On)
4	2, 3	ax-mp 5	. . . . 5 ⊢ ¬ On ∈ On
5		eleq1 2822	. . . . 5 ⊢ (ω = On → (ω ∈ On ↔ On ∈ On))
6	4, 5	mtbiri 327	. . . 4 ⊢ (ω = On → ¬ ω ∈ On)
7	6	con2i 139	. . 3 ⊢ (ω ∈ On → ¬ ω = On)
8		imnan 399	. . 3 ⊢ ((ω ∈ On → ¬ ω = On) ↔ ¬ (ω ∈ On ∧ ω = On))
9	7, 8	mpbi 230	. 2 ⊢ ¬ (ω ∈ On ∧ ω = On)
10		xor2 1518	. 2 ⊢ ((ω ∈ On ⊻ ω = On) ↔ ((ω ∈ On ∨ ω = On) ∧ ¬ (ω ∈ On ∧ ω = On)))
11	1, 9, 10	mpbir2an 711	1 ⊢ (ω ∈ On ⊻ ω = On)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ⊻ wxo 1512   = wceq 1541   ∈ wcel 2113  Vcvv 3438  Oncon0 6315  ωcom 7806

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 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1513  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-tr 5204  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-ord 6318  df-on 6319  df-lim 6320  df-om 7807

This theorem is referenced by: (None)