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Mathbox for Glauco Siliprandi |
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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 1859 | . 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 44954 | . 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
5 | df-rex 3069 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥)) | |
6 | 1, 4, 5 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 Ⅎwnfc 2888 ∃wrex 3068 ∪ cuni 4912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rex 3069 df-v 3480 df-uni 4913 |
This theorem is referenced by: smfresal 46744 smfpimbor1lem2 46755 |
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