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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 2189 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | alrimiv 1830 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)) |
3 | elisset 2674 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
4 | exim 1563 | . 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 1314 = wceq 1316 ∃wex 1453 ∈ wcel 1465 |
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 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-v 2662 |
This theorem is referenced by: snmg 3611 oprcl 3699 brm 3948 exmid01 4091 exss 4119 onintrab2im 4404 regexmidlemm 4417 dmxpid 4730 acexmidlem2 5739 frecabcl 6264 ixpm 6592 enm 6682 ssfilem 6737 fin0 6747 fin0or 6748 diffitest 6749 diffisn 6755 infm 6766 inffiexmid 6768 ctssdc 6966 omct 6970 ctssexmid 6992 exmidfodomrlemr 7026 exmidfodomrlemrALT 7027 exmidaclem 7032 caucvgsrlemasr 7566 suplocsrlempr 7583 gtso 7811 sup3exmid 8683 indstr 9356 negm 9375 fzm 9786 fzom 9909 rexfiuz 10729 r19.2uz 10733 resqrexlemgt0 10760 climuni 11030 bezoutlembi 11620 lcmgcdlem 11685 tgioo 12642 nninfall 13131 |
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