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Mathbox for Jeff Hankins |
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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 36323 | . 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 1537 ⊆ wss 3963 ∪ cuni 4912 class class class wbr 5148 ‘cfv 6563 topGenctg 17484 Fnecfne 36319 |
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 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-topgen 17490 df-fne 36320 |
This theorem is referenced by: fnetr 36334 fnessref 36340 fnemeet2 36350 fnejoin2 36352 |
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