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| Mirrors > Home > MPE Home > Th. List > intex | Structured version Visualization version GIF version | ||
| Description: The intersection of a nonempty class exists. Exercise 5 of [TakeutiZaring] p. 44 and its converse. (Contributed by NM, 13-Aug-2002.) |
| Ref | Expression |
|---|---|
| intex | ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | n0 4353 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
| 2 | intss1 4963 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥) | |
| 3 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 4 | 3 | ssex 5321 | . . . . 5 ⊢ (∩ 𝐴 ⊆ 𝑥 → ∩ 𝐴 ∈ V) |
| 5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
| 6 | 5 | exlimiv 1930 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
| 7 | 1, 6 | sylbi 217 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ∈ V) |
| 8 | vprc 5315 | . . . 4 ⊢ ¬ V ∈ V | |
| 9 | inteq 4949 | . . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
| 10 | int0 4962 | . . . . . 6 ⊢ ∩ ∅ = V | |
| 11 | 9, 10 | eqtrdi 2793 | . . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
| 12 | 11 | eleq1d 2826 | . . . 4 ⊢ (𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ V ∈ V)) |
| 13 | 8, 12 | mtbiri 327 | . . 3 ⊢ (𝐴 = ∅ → ¬ ∩ 𝐴 ∈ V) |
| 14 | 13 | necon2ai 2970 | . 2 ⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅) |
| 15 | 7, 14 | impbii 209 | 1 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 ∩ cint 4946 |
| 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-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-int 4947 |
| This theorem is referenced by: intnex 5345 intexab 5346 iinexg 5348 onint0 7811 onintrab 7816 onmindif2 7827 fival 9452 elfi2 9454 elfir 9455 dffi2 9463 elfiun 9470 fifo 9472 tz9.1c 9770 tz9.12lem1 9827 tz9.12lem3 9829 rankf 9834 cardf2 9983 cardval3 9992 cardid2 9993 cardcf 10292 cflim2 10303 intwun 10775 wuncval 10782 inttsk 10814 intgru 10854 gruina 10858 dfrtrcl2 15101 mremre 17647 mrcval 17653 asplss 21894 aspsubrg 21896 toponmre 23101 subbascn 23262 zarclsint 33871 insiga 34138 sigagenval 34141 sigagensiga 34142 dmsigagen 34145 dfon2lem8 35791 dfon2lem9 35792 bj-snmoore 37114 igenval 38068 pclvalN 39892 elrfi 42705 ismrcd1 42709 mzpval 42743 dmmzp 42744 oninfex2 43257 salgenval 46336 intsal 46345 |
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