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| Mirrors > Home > MPE Home > Th. List > uniexb | Structured version Visualization version GIF version | ||
| Description: The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.) | 
| Ref | Expression | 
|---|---|
| uniexb | ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uniexg 7761 | . 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V) | |
| 2 | uniexr 7784 | . 2 ⊢ (∪ 𝐴 ∈ V → 𝐴 ∈ V) | |
| 3 | 1, 2 | impbii 209 | 1 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∈ wcel 2107 Vcvv 3479 ∪ cuni 4906 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-pow 5364 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-in 3957 df-ss 3967 df-pw 4601 df-uni 4907 | 
| This theorem is referenced by: elpwpwel 7788 ixpexg 8963 rankuni 9904 unialeph 10142 ttukeylem1 10550 tgss2 22995 ordtbas2 23200 ordtbas 23201 ordttopon 23202 ordtopn1 23203 ordtopn2 23204 ordtrest2 23213 isref 23518 islocfin 23526 txbasex 23575 ptbasin2 23587 ordthmeolem 23810 alexsublem 24053 alexsub 24054 alexsubb 24055 ussid 24270 ordtrest2NEW 33923 brbigcup 35900 isfne 36341 isfne4 36342 isfne4b 36343 fnessref 36359 neibastop1 36361 fnejoin2 36371 prtex 38882 | 
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