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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 33690 | . 2 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) |
4 | 3 | simplbi 500 | 1 ⊢ (𝐴Fne𝐵 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ⊆ wss 3938 ∪ cuni 4840 class class class wbr 5068 ‘cfv 6357 topGenctg 16713 Fnecfne 33686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-topgen 16719 df-fne 33687 |
This theorem is referenced by: fnetr 33701 fnessref 33707 fnemeet2 33717 fnejoin2 33719 |
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