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Mirrors > Home > ILE Home > Th. List > eluni2 | GIF version |
Description: Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.) |
Ref | Expression |
---|---|
eluni2 | ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exancom 1572 | . 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥)) | |
2 | eluni 3709 | . 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) | |
3 | df-rex 2399 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥)) | |
4 | 1, 2, 3 | 3bitr4i 211 | 1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1453 ∈ wcel 1465 ∃wrex 2394 ∪ cuni 3706 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-v 2662 df-uni 3707 |
This theorem is referenced by: uni0b 3731 intssunim 3763 iuncom4 3790 inuni 4050 ssorduni 4373 unon 4397 cnvuni 4695 chfnrn 5499 isbasis3g 12140 eltg2b 12150 tgcl 12160 epttop 12186 txuni2 12352 |
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