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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 2242 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | alrimiv 1867 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
3 | elisset 2744 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
4 | exim 1592 | . 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 1346 = wceq 1348 ∃wex 1485 ∈ wcel 2141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-v 2732 |
This theorem is referenced by: snmg 3701 oprcl 3789 brm 4039 ss1o0el1 4183 exss 4212 onintrab2im 4502 regexmidlemm 4516 dmxpid 4832 acexmidlem2 5850 frecabcl 6378 ixpm 6708 enm 6798 ssfilem 6853 fin0 6863 fin0or 6864 diffitest 6865 diffisn 6871 infm 6882 inffiexmid 6884 ctssdc 7090 omct 7094 ctssexmid 7126 exmidfodomrlemr 7179 exmidfodomrlemrALT 7180 exmidaclem 7185 caucvgsrlemasr 7752 suplocsrlempr 7769 gtso 7998 sup3exmid 8873 indstr 9552 negm 9574 fzm 9994 fzom 10120 rexfiuz 10953 r19.2uz 10957 resqrexlemgt0 10984 climuni 11256 bezoutlembi 11960 lcmgcdlem 12031 pcprecl 12243 pc2dvds 12283 nninfdclemcl 12403 tgioo 13340 pw1nct 14036 nninfall 14042 |
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