Theorem xoromon 35270

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 3456	. . . . . 6 ⊢ (¬ On ∈ V → ¬ On ∈ On)
4	2, 3	ax-mp 5	. . . . 5 ⊢ ¬ On ∈ On
5		eleq1 2827	. . . . 5 ⊢ (ω = On → (ω ∈ On ↔ On ∈ On))
6	4, 5	mtbiri 328	. . . 4 ⊢ (ω = On → ¬ ω ∈ On)
7	6	con2i 139	. . 3 ⊢ (ω ∈ On → ¬ ω = On)
8		imnan 400	. . 3 ⊢ ((ω ∈ On → ¬ ω = On) ↔ ¬ (ω ∈ On ∧ ω = On))
9	7, 8	mpbi 231	. 2 ⊢ ¬ (ω ∈ On ∧ ω = On)
10		xor2 1524	. 2 ⊢ ((ω ∈ On ⊻ ω = On) ↔ ((ω ∈ On ∨ ω = On) ∧ ¬ (ω ∈ On ∧ ω = On)))
11	1, 9, 10	mpbir2an 717	1 ⊢ (ω ∈ On ⊻ ω = On)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 853   ⊻ wxo 1518   = wceq 1547   ∈ wcel 2119  Vcvv 3431  Oncon0 6310  ωcom 7806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-xor 1519  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-tr 5180  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-ord 6313  df-on 6314  df-lim 6315  df-om 7807

This theorem is referenced by: (None)