Theorem xoromon 35421

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 7873. (Contributed by BTernaryTau, 25-Jan-2026.)

Assertion

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

Proof of Theorem xoromon

Step	Hyp	Ref	Expression
1		omon 7873	. 2 ⊢ (ω ∈ On ∨ ω = On)
2		onprc 7776	. . . . . 6 ⊢ ¬ On ∈ V
3		prcnel 3488	. . . . . 6 ⊢ (¬ On ∈ V → ¬ On ∈ On)
4	2, 3	ax-mp 5	. . . . 5 ⊢ ¬ On ∈ On
5		eleq1 2857	. . . . 5 ⊢ (ω = On → (ω ∈ On ↔ On ∈ On))
6	4, 5	mtbiri 330	. . . 4 ⊢ (ω = On → ¬ ω ∈ On)
7	6	con2i 140	. . 3 ⊢ (ω ∈ On → ¬ ω = On)
8		imnan 404	. . 3 ⊢ ((ω ∈ On → ¬ ω = On) ↔ ¬ (ω ∈ On ∧ ω = On))
9	7, 8	mpbi 233	. 2 ⊢ ¬ (ω ∈ On ∧ ω = On)
10		xor2 1544	. 2 ⊢ ((ω ∈ On ⊻ ω = On) ↔ ((ω ∈ On ∨ ω = On) ∧ ¬ (ω ∈ On ∧ ω = On)))
11	1, 9, 10	mpbir2an 723	1 ⊢ (ω ∈ On ⊻ ω = On)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   ⊻ wxo 1538   = wceq 1567   ∈ wcel 2149  Vcvv 3463  Oncon0 6361  ωcom 7861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-xor 1539  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365  df-lim 6366  df-om 7862

This theorem is referenced by: (None)