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Mirrors > Home > MPE Home > Th. List > Mathboxes > isfne2 | Structured version Visualization version GIF version |
Description: The predicate "𝐵 is finer than 𝐴". (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
isfne.1 | ⊢ 𝑋 = ∪ 𝐴 |
isfne.2 | ⊢ 𝑌 = ∪ 𝐵 |
Ref | Expression |
---|---|
isfne2 | ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfne.1 | . . 3 ⊢ 𝑋 = ∪ 𝐴 | |
2 | isfne.2 | . . 3 ⊢ 𝑌 = ∪ 𝐵 | |
3 | 1, 2 | isfne4 33801 | . 2 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) |
4 | dfss3 3903 | . . . 4 ⊢ (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵)) | |
5 | eltg2b 21564 | . . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) | |
6 | 5 | ralbidv 3162 | . . . 4 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) |
7 | 4, 6 | syl5bb 286 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) |
8 | 7 | anbi2d 631 | . 2 ⊢ (𝐵 ∈ 𝐶 → ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))) |
9 | 3, 8 | syl5bb 286 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 ∪ cuni 4800 class class class wbr 5030 ‘cfv 6324 topGenctg 16703 Fnecfne 33797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-topgen 16709 df-fne 33798 |
This theorem is referenced by: fness 33810 fneref 33811 fnessref 33818 |
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