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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fnebas | Structured version Visualization version GIF version | ||
| Description: A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.) | 
| Ref | Expression | 
|---|---|
| fnebas.1 | ⊢ 𝑋 = ∪ 𝐴 | 
| fnebas.2 | ⊢ 𝑌 = ∪ 𝐵 | 
| Ref | Expression | 
|---|---|
| fnebas | ⊢ (𝐴Fne𝐵 → 𝑋 = 𝑌) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fnebas.1 | . . 3 ⊢ 𝑋 = ∪ 𝐴 | |
| 2 | fnebas.2 | . . 3 ⊢ 𝑌 = ∪ 𝐵 | |
| 3 | 1, 2 | isfne4 36341 | . 2 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) | 
| 4 | 3 | simplbi 497 | 1 ⊢ (𝐴Fne𝐵 → 𝑋 = 𝑌) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 ‘cfv 6561 topGenctg 17482 Fnecfne 36337 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-topgen 17488 df-fne 36338 | 
| This theorem is referenced by: fnetr 36352 fnessref 36358 fnemeet2 36368 fnejoin2 36370 | 
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