Theorem xoromon 35384

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 7858. (Contributed by BTernaryTau, 25-Jan-2026.)

Assertion

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

Proof of Theorem xoromon

Step	Hyp	Ref	Expression
1		omon 7858	. 2 ⊢ (ω ∈ On ∨ ω = On)
2		onprc 7761	. . . . . 6 ⊢ ¬ On ∈ V
3		prcnel 3479	. . . . . 6 ⊢ (¬ On ∈ V → ¬ On ∈ On)
4	2, 3	ax-mp 5	. . . . 5 ⊢ ¬ On ∈ On
5		eleq1 2850	. . . . 5 ⊢ (ω = On → (ω ∈ On ↔ On ∈ On))
6	4, 5	mtbiri 329	. . . 4 ⊢ (ω = On → ¬ ω ∈ On)
7	6	con2i 139	. . 3 ⊢ (ω ∈ On → ¬ ω = On)
8		imnan 403	. . 3 ⊢ ((ω ∈ On → ¬ ω = On) ↔ ¬ (ω ∈ On ∧ ω = On))
9	7, 8	mpbi 232	. 2 ⊢ ¬ (ω ∈ On ∧ ω = On)
10		xor2 1537	. 2 ⊢ ((ω ∈ On ⊻ ω = On) ↔ ((ω ∈ On ∨ ω = On) ∧ ¬ (ω ∈ On ∧ ω = On)))
11	1, 9, 10	mpbir2an 721	1 ⊢ (ω ∈ On ⊻ ω = On)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   ⊻ wxo 1531   = wceq 1560   ∈ wcel 2142  Vcvv 3454  Oncon0 6346  ωcom 7846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-xor 1532  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6349  df-on 6350  df-lim 6351  df-om 7847

This theorem is referenced by: (None)