Metamath Proof Explorer


Theorem ficardadju

Description: The disjoint union of finite sets is equinumerous to the ordinal sum of the cardinalities of those sets. (Contributed by BTernaryTau, 3-Jul-2024)

Ref Expression
Assertion ficardadju ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( 𝐴𝐵 ) ≈ ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 ficardom ( 𝐴 ∈ Fin → ( card ‘ 𝐴 ) ∈ ω )
2 ficardom ( 𝐵 ∈ Fin → ( card ‘ 𝐵 ) ∈ ω )
3 nnadju ( ( ( card ‘ 𝐴 ) ∈ ω ∧ ( card ‘ 𝐵 ) ∈ ω ) → ( card ‘ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ) = ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) )
4 df-dju ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) = ( ( { ∅ } × ( card ‘ 𝐴 ) ) ∪ ( { 1o } × ( card ‘ 𝐵 ) ) )
5 snfi { ∅ } ∈ Fin
6 nnfi ( ( card ‘ 𝐴 ) ∈ ω → ( card ‘ 𝐴 ) ∈ Fin )
7 xpfi ( ( { ∅ } ∈ Fin ∧ ( card ‘ 𝐴 ) ∈ Fin ) → ( { ∅ } × ( card ‘ 𝐴 ) ) ∈ Fin )
8 5 6 7 sylancr ( ( card ‘ 𝐴 ) ∈ ω → ( { ∅ } × ( card ‘ 𝐴 ) ) ∈ Fin )
9 snfi { 1o } ∈ Fin
10 nnfi ( ( card ‘ 𝐵 ) ∈ ω → ( card ‘ 𝐵 ) ∈ Fin )
11 xpfi ( ( { 1o } ∈ Fin ∧ ( card ‘ 𝐵 ) ∈ Fin ) → ( { 1o } × ( card ‘ 𝐵 ) ) ∈ Fin )
12 9 10 11 sylancr ( ( card ‘ 𝐵 ) ∈ ω → ( { 1o } × ( card ‘ 𝐵 ) ) ∈ Fin )
13 unfi ( ( ( { ∅ } × ( card ‘ 𝐴 ) ) ∈ Fin ∧ ( { 1o } × ( card ‘ 𝐵 ) ) ∈ Fin ) → ( ( { ∅ } × ( card ‘ 𝐴 ) ) ∪ ( { 1o } × ( card ‘ 𝐵 ) ) ) ∈ Fin )
14 8 12 13 syl2an ( ( ( card ‘ 𝐴 ) ∈ ω ∧ ( card ‘ 𝐵 ) ∈ ω ) → ( ( { ∅ } × ( card ‘ 𝐴 ) ) ∪ ( { 1o } × ( card ‘ 𝐵 ) ) ) ∈ Fin )
15 4 14 eqeltrid ( ( ( card ‘ 𝐴 ) ∈ ω ∧ ( card ‘ 𝐵 ) ∈ ω ) → ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ∈ Fin )
16 ficardid ( ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ∈ Fin → ( card ‘ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ) ≈ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) )
17 15 16 syl ( ( ( card ‘ 𝐴 ) ∈ ω ∧ ( card ‘ 𝐵 ) ∈ ω ) → ( card ‘ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ) ≈ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) )
18 3 17 eqbrtrrd ( ( ( card ‘ 𝐴 ) ∈ ω ∧ ( card ‘ 𝐵 ) ∈ ω ) → ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) ≈ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) )
19 1 2 18 syl2an ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) ≈ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) )
20 ficardid ( 𝐴 ∈ Fin → ( card ‘ 𝐴 ) ≈ 𝐴 )
21 ficardid ( 𝐵 ∈ Fin → ( card ‘ 𝐵 ) ≈ 𝐵 )
22 djuen ( ( ( card ‘ 𝐴 ) ≈ 𝐴 ∧ ( card ‘ 𝐵 ) ≈ 𝐵 ) → ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ≈ ( 𝐴𝐵 ) )
23 20 21 22 syl2an ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ≈ ( 𝐴𝐵 ) )
24 entr ( ( ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) ≈ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ∧ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ≈ ( 𝐴𝐵 ) ) → ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) ≈ ( 𝐴𝐵 ) )
25 19 23 24 syl2anc ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) ≈ ( 𝐴𝐵 ) )
26 25 ensymd ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( 𝐴𝐵 ) ≈ ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) )