This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem nnadjuALT

Description: Shorter proof of nnadju using ax-rep . (Contributed by Paul Chapman, 11-Apr-2009) (Revised by Mario Carneiro, 6-Feb-2013) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion nnadjuALT ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( card ‘ ( 𝐴 ⊔ 𝐵 ) ) = ( 𝐴 +o 𝐵 ) )

Proof

Step Hyp Ref Expression
1 nnon ⊢ ( 𝐴 ∈ ω → 𝐴 ∈ On )
2 nnon ⊢ ( 𝐵 ∈ ω → 𝐵 ∈ On )
3 onadju ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o 𝐵 ) ≈ ( 𝐴 ⊔ 𝐵 ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 +o 𝐵 ) ≈ ( 𝐴 ⊔ 𝐵 ) )
5 carden2b ⊢ ( ( 𝐴 +o 𝐵 ) ≈ ( 𝐴 ⊔ 𝐵 ) → ( card ‘ ( 𝐴 +o 𝐵 ) ) = ( card ‘ ( 𝐴 ⊔ 𝐵 ) ) )
6 4 5 syl ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( card ‘ ( 𝐴 +o 𝐵 ) ) = ( card ‘ ( 𝐴 ⊔ 𝐵 ) ) )
7 nnacl ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 +o 𝐵 ) ∈ ω )
8 cardnn ⊢ ( ( 𝐴 +o 𝐵 ) ∈ ω → ( card ‘ ( 𝐴 +o 𝐵 ) ) = ( 𝐴 +o 𝐵 ) )
9 7 8 syl ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( card ‘ ( 𝐴 +o 𝐵 ) ) = ( 𝐴 +o 𝐵 ) )
10 6 9 eqtr3d ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( card ‘ ( 𝐴 ⊔ 𝐵 ) ) = ( 𝐴 +o 𝐵 ) )