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Mirrors > Home > MPE Home > Th. List > elunirab | Structured version Visualization version GIF version |
Description: Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.) |
Ref | Expression |
---|---|
elunirab | ⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluniab 4447 | . 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑))) | |
2 | df-rab 2921 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
3 | 2 | unieqi 4445 | . . 3 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} = ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} |
4 | 3 | eleq2i 2693 | . 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
5 | df-rex 2918 | . . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝐴 ∈ 𝑥 ∧ 𝜑))) | |
6 | an12 838 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝐴 ∈ 𝑥 ∧ 𝜑)) ↔ (𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑))) | |
7 | 6 | exbii 1774 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝐴 ∈ 𝑥 ∧ 𝜑)) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
8 | 5, 7 | bitri 264 | . 2 ⊢ (∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
9 | 1, 4, 8 | 3bitr4i 292 | 1 ⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∃wex 1704 ∈ wcel 1990 {cab 2608 ∃wrex 2913 {crab 2916 ∪ cuni 4436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-uni 4437 |
This theorem is referenced by: neiptopuni 20934 cmpcov2 21193 tgcmp 21204 hauscmplem 21209 conncompid 21234 alexsubALT 21855 cvmliftlem15 31280 fnessref 32352 cover2 33508 |
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