Theorem 1sdom2

Description: Ordinal 1 is strictly dominated by ordinal 2. For a shorter proof requiring ax-un , see 1sdom2ALT . (Contributed by NM, 4-Apr-2007) Avoid ax-un . (Revised by BTernaryTau, 8-Dec-2024)

	Ref	Expression
Assertion	1sdom2	⊢ 1o ≺ 2o

Proof

Step	Hyp	Ref	Expression
1		2on0	⊢ 2o ≠ ∅
2		2oex	⊢ 2o ∈ V
3	2	0sdom	⊢ ( ∅ ≺ 2o ↔ 2o ≠ ∅ )
4	1 3	mpbir	⊢ ∅ ≺ 2o
5		0sdom1dom	⊢ ( ∅ ≺ 2o ↔ 1o ≼ 2o )
6	4 5	mpbi	⊢ 1o ≼ 2o
7		snnen2o	⊢ ¬ { ∅ } ≈ 2o
8		df1o2	⊢ 1o = { ∅ }
9	8	breq1i	⊢ ( 1o ≈ 2o ↔ { ∅ } ≈ 2o )
10	7 9	mtbir	⊢ ¬ 1o ≈ 2o
11		brsdom	⊢ ( 1o ≺ 2o ↔ ( 1o ≼ 2o ∧ ¬ 1o ≈ 2o ) )
12	6 10 11	mpbir2an	⊢ 1o ≺ 2o