Theorem xoromon 35594

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 7875. (Contributed by BTernaryTau, 25-Jan-2026.)

Assertion

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

Proof of Theorem xoromon

Step	Hyp	Ref	Expression
1		omon 7875	. 2 ⊢ (ω ∈ On ∨ ω = On)
2		onprc 7778	. . . . . 6 ⊢ ¬ On ∈ V
3		prcnel 3475	. . . . . 6 ⊢ (¬ On ∈ V → ¬ On ∈ On)
4	2, 3	ax-mp 5	. . . . 5 ⊢ ¬ On ∈ On
5		eleq1 2848	. . . . 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 405	. . 3 ⊢ ((ω ∈ On → ¬ ω = On) ↔ ¬ (ω ∈ On ∧ ω = On))
9	7, 8	mpbi 233	. 2 ⊢ ¬ (ω ∈ On ∧ ω = On)
10		xor2 1547	. 2 ⊢ ((ω ∈ On ⊻ ω = On) ↔ ((ω ∈ On ∨ ω = On) ∧ ¬ (ω ∈ On ∧ ω = On)))
11	1, 9, 10	mpbir2an 724	1 ⊢ (ω ∈ On ⊻ ω = On)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ⊻ wxo 1541   = wceq 1570   ∈ wcel 2145  Vcvv 3450  Oncon0 6357  ωcom 7863

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361  df-lim 6362  df-om 7864

This theorem is used by: (None)