Metamath Proof Explorer


Theorem onadju

Description: The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013) (Revised by Jim Kingdon, 7-Sep-2023)

Ref Expression
Assertion onadju ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o 𝐵 ) ≈ ( 𝐴𝐵 ) )

Proof

Step Hyp Ref Expression
1 enrefg ( 𝐴 ∈ On → 𝐴𝐴 )
2 1 adantr ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐴𝐴 )
3 simpr ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐵 ∈ On )
4 eqid ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) = ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) )
5 4 oacomf1olem ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : 𝐵1-1-onto→ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∧ ( ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∩ 𝐴 ) = ∅ ) )
6 5 ancoms ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : 𝐵1-1-onto→ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∧ ( ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∩ 𝐴 ) = ∅ ) )
7 6 simpld ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : 𝐵1-1-onto→ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) )
8 f1oeng ( ( 𝐵 ∈ On ∧ ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : 𝐵1-1-onto→ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) → 𝐵 ≈ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) )
9 3 7 8 syl2anc ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐵 ≈ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) )
10 incom ( 𝐴 ∩ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) = ( ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∩ 𝐴 )
11 6 simprd ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∩ 𝐴 ) = ∅ )
12 10 11 eqtrid ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ∩ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) = ∅ )
13 djuenun ( ( 𝐴𝐴𝐵 ≈ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∧ ( 𝐴 ∩ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) = ∅ ) → ( 𝐴𝐵 ) ≈ ( 𝐴 ∪ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) )
14 2 9 12 13 syl3anc ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴𝐵 ) ≈ ( 𝐴 ∪ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) )
15 oarec ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o 𝐵 ) = ( 𝐴 ∪ ran ( 𝑥𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) )
16 14 15 breqtrrd ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴𝐵 ) ≈ ( 𝐴 +o 𝐵 ) )
17 16 ensymd ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o 𝐵 ) ≈ ( 𝐴𝐵 ) )