Theorem xoromon 35835

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 7923. (Contributed by BTernaryTau, 25-Jan-2026.)

Assertion

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

Proof of Theorem xoromon

Step	Hyp	Ref	Expression
1		omon 7923	. 2 ⊢ (ω ∈ On ∨ ω = On)
2		onprc 7824	. . . . . 6 ⊢ ¬ On ∈ V
3		prcnel 3479	. . . . . 6 ⊢ (¬ On ∈ V → ¬ On ∈ On)
4	2, 3	ax-mp 5	. . . . 5 ⊢ ¬ On ∈ On
5		eleq1 2852	. . . . 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 2147  Vcvv 3454  Oncon0 6388  ωcom 7911

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392  df-lim 6393  df-om 7912

This theorem is used by: (None)