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 e. ( card ` B ) -> -. A ~~ B )`

Proof

Step Hyp Ref Expression
1 elfvdm
` |-  ( A e. ( card ` B ) -> B e. dom card )`
2 cardon
` |-  ( card ` B ) e. On`
3 2 oneli
` |-  ( A e. ( card ` B ) -> A e. On )`
4 breq1
` |-  ( x = A -> ( x ~~ B <-> A ~~ B ) )`
5 4 onintss
` |-  ( A e. On -> ( A ~~ B -> |^| { x e. On | x ~~ B } C_ A ) )`
6 3 5 syl
` |-  ( A e. ( card ` B ) -> ( A ~~ B -> |^| { x e. On | x ~~ B } C_ A ) )`
` |-  ( ( B e. dom card /\ A e. ( card ` B ) ) -> ( A ~~ B -> |^| { x e. On | x ~~ B } C_ A ) )`
8 cardval3
` |-  ( B e. dom card -> ( card ` B ) = |^| { x e. On | x ~~ B } )`
9 8 sseq1d
` |-  ( B e. dom card -> ( ( card ` B ) C_ A <-> |^| { x e. On | x ~~ B } C_ A ) )`
` |-  ( ( B e. dom card /\ A e. ( card ` B ) ) -> ( ( card ` B ) C_ A <-> |^| { x e. On | x ~~ B } C_ A ) )`
11 7 10 sylibrd
` |-  ( ( B e. dom card /\ A e. ( card ` B ) ) -> ( A ~~ B -> ( card ` B ) C_ A ) )`
12 ontri1
` |-  ( ( ( card ` B ) e. On /\ A e. On ) -> ( ( card ` B ) C_ A <-> -. A e. ( card ` B ) ) )`
13 2 3 12 sylancr
` |-  ( A e. ( card ` B ) -> ( ( card ` B ) C_ A <-> -. A e. ( card ` B ) ) )`
` |-  ( ( B e. dom card /\ A e. ( card ` B ) ) -> ( ( card ` B ) C_ A <-> -. A e. ( card ` B ) ) )`
` |-  ( ( B e. dom card /\ A e. ( card ` B ) ) -> ( A ~~ B -> -. A e. ( card ` B ) ) )`
` |-  ( ( B e. dom card /\ A e. ( card ` B ) ) -> ( A e. ( card ` B ) -> -. A ~~ B ) )`
` |-  ( B e. dom card -> ( A e. ( card ` B ) -> ( A e. ( card ` B ) -> -. A ~~ B ) ) )`
` |-  ( B e. dom card -> ( A e. ( card ` B ) -> -. A ~~ B ) )`
` |-  ( A e. ( card ` B ) -> -. A ~~ B )`