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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 ) ≈ ( 𝐴 ⊔ 𝐵 ) )

Proof

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