New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > eqsstri | GIF version |
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.) |
Ref | Expression |
---|---|
eqsstr.1 | ⊢ A = B |
eqsstr.2 | ⊢ B ⊆ C |
Ref | Expression |
---|---|
eqsstri | ⊢ A ⊆ C |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsstr.2 | . 2 ⊢ B ⊆ C | |
2 | eqsstr.1 | . . 3 ⊢ A = B | |
3 | 2 | sseq1i 3295 | . 2 ⊢ (A ⊆ C ↔ B ⊆ C) |
4 | 1, 3 | mpbir 200 | 1 ⊢ A ⊆ C |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ⊆ wss 3257 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 |
This theorem is referenced by: eqsstr3i 3302 ssrab2 3351 rabssab 3352 difsscompl 3549 pw1ss1c 4158 pw1sspw 4171 opkabssvvki 4209 imagekrelk 4273 dmopabss 4916 resss 4988 rnin 5037 rnxpss 5053 fun0 5154 fnres 5199 f0 5248 fvopab4ndm 5390 ffvresb 5431 isoini2 5498 dmoprabss 5575 dmmptss 5685 ecss 5966 nenpw1pwlem2 6085 sbthlem1 6203 spacssnc 6284 frecxp 6314 |
Copyright terms: Public domain | W3C validator |