Theorem xoromon 35232

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 7829. (Contributed by BTernaryTau, 25-Jan-2026.)

Assertion

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

Proof of Theorem xoromon

Step	Hyp	Ref	Expression
1		omon 7829	. 2 ⊢ (ω ∈ On ∨ ω = On)
2		onprc 7732	. . . . . 6 ⊢ ¬ On ∈ V
3		prcnel 3455	. . . . . 6 ⊢ (¬ On ∈ V → ¬ On ∈ On)
4	2, 3	ax-mp 5	. . . . 5 ⊢ ¬ On ∈ On
5		eleq1 2824	. . . . 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 1519	. 2 ⊢ ((ω ∈ On ⊻ ω = On) ↔ ((ω ∈ On ∨ ω = On) ∧ ¬ (ω ∈ On ∧ ω = On)))
11	1, 9, 10	mpbir2an 712	1 ⊢ (ω ∈ On ⊻ ω = On)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   ⊻ wxo 1513   = wceq 1542   ∈ wcel 2114  Vcvv 3429  Oncon0 6323  ωcom 7817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1514  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327  df-lim 6328  df-om 7818

This theorem is referenced by: (None)