Description: Any set is equinumerous to its cardinal number. Closed theorem form of cardid . (Contributed by David Moews, 1-May-2017)
Ref | Expression | ||
---|---|---|---|
Assertion | cardidg | ⊢ ( 𝐴 ∈ 𝐵 → ( card ‘ 𝐴 ) ≈ 𝐴 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex | ⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V ) | |
2 | cardeqv | ⊢ dom card = V | |
3 | 2 | eleq2i | ⊢ ( 𝐴 ∈ dom card ↔ 𝐴 ∈ V ) |
4 | cardid2 | ⊢ ( 𝐴 ∈ dom card → ( card ‘ 𝐴 ) ≈ 𝐴 ) | |
5 | 3 4 | sylbir | ⊢ ( 𝐴 ∈ V → ( card ‘ 𝐴 ) ≈ 𝐴 ) |
6 | 1 5 | syl | ⊢ ( 𝐴 ∈ 𝐵 → ( card ‘ 𝐴 ) ≈ 𝐴 ) |