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Mirrors > Home > MPE Home > Th. List > elfir | Structured version Visualization version Unicode version |
Description: Sufficient condition for an element of . (Contributed by Mario Carneiro, 24-Nov-2013.) |
Ref | Expression |
---|---|
elfir |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1061 | . . . . . 6 | |
2 | elpw2g 4827 | . . . . . 6 | |
3 | 1, 2 | syl5ibr 236 | . . . . 5 |
4 | 3 | imp 445 | . . . 4 |
5 | simpr3 1069 | . . . 4 | |
6 | 4, 5 | elind 3798 | . . 3 |
7 | eqid 2622 | . . 3 | |
8 | inteq 4478 | . . . . 5 | |
9 | 8 | eqeq2d 2632 | . . . 4 |
10 | 9 | rspcev 3309 | . . 3 |
11 | 6, 7, 10 | sylancl 694 | . 2 |
12 | simp2 1062 | . . . 4 | |
13 | intex 4820 | . . . 4 | |
14 | 12, 13 | sylib 208 | . . 3 |
15 | id 22 | . . 3 | |
16 | elfi 8319 | . . 3 | |
17 | 14, 15, 16 | syl2anr 495 | . 2 |
18 | 11, 17 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wrex 2913 cvv 3200 cin 3573 wss 3574 c0 3915 cpw 4158 cint 4475 cfv 5888 cfn 7955 cfi 8316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-fi 8317 |
This theorem is referenced by: intrnfi 8322 ssfii 8325 elfiun 8336 ptbasfi 21384 fbssint 21642 filintn0 21665 alexsublem 21848 ispisys2 30216 |
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