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Mirrors > Home > ILE Home > Th. List > elex2 | GIF version |
Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) |
Ref | Expression |
---|---|
elex2 | ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1a 2249 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | alrimiv 1874 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
3 | elisset 2753 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
4 | exim 1599 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥 ∈ 𝐵)) | |
5 | 2, 3, 4 | sylc 62 | 1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1351 = wceq 1353 ∃wex 1492 ∈ wcel 2148 |
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 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-v 2741 |
This theorem is referenced by: snmg 3712 oprcl 3804 brm 4055 ss1o0el1 4199 exss 4229 onintrab2im 4519 regexmidlemm 4533 dmxpid 4850 acexmidlem2 5874 frecabcl 6402 ixpm 6732 enm 6822 ssfilem 6877 fin0 6887 fin0or 6888 diffitest 6889 diffisn 6895 infm 6906 inffiexmid 6908 ctssdc 7114 omct 7118 ctssexmid 7150 exmidfodomrlemr 7203 exmidfodomrlemrALT 7204 exmidaclem 7209 caucvgsrlemasr 7791 suplocsrlempr 7808 gtso 8038 sup3exmid 8916 indstr 9595 negm 9617 fzm 10040 fzom 10166 rexfiuz 11000 r19.2uz 11004 resqrexlemgt0 11031 climuni 11303 bezoutlembi 12008 lcmgcdlem 12079 pcprecl 12291 pc2dvds 12331 nninfdclemcl 12451 dfgrp3m 12974 issubg2m 13054 issubgrpd2 13055 issubg3 13057 issubg4m 13058 grpissubg 13059 subgintm 13063 nmzsubg 13075 dvdsr02 13279 01eq0ring 13335 subrgugrp 13366 lmodfopnelem1 13419 rmodislmodlem 13445 rmodislmod 13446 lss1 13454 lsssubg 13469 islss3 13471 islss4 13474 lss1d 13475 lssintclm 13476 cnsubglem 13512 tgioo 14085 pw1nct 14791 nninfall 14797 |
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