# Metamath Proof Explorer

## Theorem cardne

Description: No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of TakeutiZaring p. 85. (Contributed by Mario Carneiro, 9-Jan-2013)

Ref Expression
Assertion cardne ( 𝐴 ∈ ( card ‘ 𝐵 ) → ¬ 𝐴𝐵 )

### Proof

Step Hyp Ref Expression
1 elfvdm ( 𝐴 ∈ ( card ‘ 𝐵 ) → 𝐵 ∈ dom card )
2 cardon ( card ‘ 𝐵 ) ∈ On
3 2 oneli ( 𝐴 ∈ ( card ‘ 𝐵 ) → 𝐴 ∈ On )
4 breq1 ( 𝑥 = 𝐴 → ( 𝑥𝐵𝐴𝐵 ) )
5 4 onintss ( 𝐴 ∈ On → ( 𝐴𝐵 { 𝑥 ∈ On ∣ 𝑥𝐵 } ⊆ 𝐴 ) )
6 3 5 syl ( 𝐴 ∈ ( card ‘ 𝐵 ) → ( 𝐴𝐵 { 𝑥 ∈ On ∣ 𝑥𝐵 } ⊆ 𝐴 ) )
7 6 adantl ( ( 𝐵 ∈ dom card ∧ 𝐴 ∈ ( card ‘ 𝐵 ) ) → ( 𝐴𝐵 { 𝑥 ∈ On ∣ 𝑥𝐵 } ⊆ 𝐴 ) )
8 cardval3 ( 𝐵 ∈ dom card → ( card ‘ 𝐵 ) = { 𝑥 ∈ On ∣ 𝑥𝐵 } )
9 8 sseq1d ( 𝐵 ∈ dom card → ( ( card ‘ 𝐵 ) ⊆ 𝐴 { 𝑥 ∈ On ∣ 𝑥𝐵 } ⊆ 𝐴 ) )
10 9 adantr ( ( 𝐵 ∈ dom card ∧ 𝐴 ∈ ( card ‘ 𝐵 ) ) → ( ( card ‘ 𝐵 ) ⊆ 𝐴 { 𝑥 ∈ On ∣ 𝑥𝐵 } ⊆ 𝐴 ) )
11 7 10 sylibrd ( ( 𝐵 ∈ dom card ∧ 𝐴 ∈ ( card ‘ 𝐵 ) ) → ( 𝐴𝐵 → ( card ‘ 𝐵 ) ⊆ 𝐴 ) )
12 ontri1 ( ( ( card ‘ 𝐵 ) ∈ On ∧ 𝐴 ∈ On ) → ( ( card ‘ 𝐵 ) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ( card ‘ 𝐵 ) ) )
13 2 3 12 sylancr ( 𝐴 ∈ ( card ‘ 𝐵 ) → ( ( card ‘ 𝐵 ) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ( card ‘ 𝐵 ) ) )
14 13 adantl ( ( 𝐵 ∈ dom card ∧ 𝐴 ∈ ( card ‘ 𝐵 ) ) → ( ( card ‘ 𝐵 ) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ( card ‘ 𝐵 ) ) )
15 11 14 sylibd ( ( 𝐵 ∈ dom card ∧ 𝐴 ∈ ( card ‘ 𝐵 ) ) → ( 𝐴𝐵 → ¬ 𝐴 ∈ ( card ‘ 𝐵 ) ) )
16 15 con2d ( ( 𝐵 ∈ dom card ∧ 𝐴 ∈ ( card ‘ 𝐵 ) ) → ( 𝐴 ∈ ( card ‘ 𝐵 ) → ¬ 𝐴𝐵 ) )
17 16 ex ( 𝐵 ∈ dom card → ( 𝐴 ∈ ( card ‘ 𝐵 ) → ( 𝐴 ∈ ( card ‘ 𝐵 ) → ¬ 𝐴𝐵 ) ) )
18 17 pm2.43d ( 𝐵 ∈ dom card → ( 𝐴 ∈ ( card ‘ 𝐵 ) → ¬ 𝐴𝐵 ) )
19 1 18 mpcom ( 𝐴 ∈ ( card ‘ 𝐵 ) → ¬ 𝐴𝐵 )