Theorem bdss 15594

Description: The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.)

Hypothesis

Ref	Expression
bdss.1	|- BOUNDED A

Assertion

Ref	Expression
bdss	|- BOUNDED x C_ A

Proof of Theorem bdss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdss.1	. . . 4 |- BOUNDED A
2	1	bdeli 15576	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 15548	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3173	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 15555	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 2167   A.wral 2475   C_ wss 3157  BOUNDED wbd 15542  BOUNDED wbdc 15570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-bd0 15543  ax-bdal 15548

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-in 3163  df-ss 3170  df-bdc 15571

This theorem is referenced by:  bdeq0  15597  bdcpw  15599  bdvsn  15604  bdop  15605  bdeqsuc  15611  bj-nntrans  15681  bj-omtrans  15686