Description: Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of TakeutiZaring p. 77. (Contributed by NM, 22-Sep-2003)
Ref | Expression | ||
---|---|---|---|
Assertion | r1ord2 | |- ( B e. On -> ( A e. B -> ( R1 ` A ) C_ ( R1 ` B ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1tr | |- Tr ( R1 ` B ) |
|
2 | r1ord | |- ( B e. On -> ( A e. B -> ( R1 ` A ) e. ( R1 ` B ) ) ) |
|
3 | trss | |- ( Tr ( R1 ` B ) -> ( ( R1 ` A ) e. ( R1 ` B ) -> ( R1 ` A ) C_ ( R1 ` B ) ) ) |
|
4 | 1 2 3 | mpsylsyld | |- ( B e. On -> ( A e. B -> ( R1 ` A ) C_ ( R1 ` B ) ) ) |