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Mirrors > Home > MPE Home > Th. List > Mathboxes > eluni2f | Structured version Visualization version GIF version |
Description: Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
eluni2f.1 | ⊢ Ⅎ𝑥𝐴 |
eluni2f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
eluni2f | ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exancom 1852 | . 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥)) | |
2 | eluni2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | eluni2f.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | elunif 41150 | . 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
5 | df-rex 3141 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥)) | |
6 | 1, 4, 5 | 3bitr4i 304 | 1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1771 ∈ wcel 2105 Ⅎwnfc 2958 ∃wrex 3136 ∪ cuni 4830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-v 3494 df-uni 4831 |
This theorem is referenced by: smfresal 42940 smfpimbor1lem2 42951 |
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