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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ispointN | Structured version Visualization version GIF version | ||
| Description: The predicate "is a point". (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| ispoint.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| ispoint.p | ⊢ 𝑃 = (Points‘𝐾) |
| Ref | Expression |
|---|---|
| ispointN | ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ispoint.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 2 | ispoint.p | . . . 4 ⊢ 𝑃 = (Points‘𝐾) | |
| 3 | 1, 2 | pointsetN 39743 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑃 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}}) |
| 4 | 3 | eleq2d 2827 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}})) |
| 5 | vsnex 5434 | . . . . 5 ⊢ {𝑎} ∈ V | |
| 6 | eleq1 2829 | . . . . 5 ⊢ (𝑋 = {𝑎} → (𝑋 ∈ V ↔ {𝑎} ∈ V)) | |
| 7 | 5, 6 | mpbiri 258 | . . . 4 ⊢ (𝑋 = {𝑎} → 𝑋 ∈ V) |
| 8 | 7 | rexlimivw 3151 | . . 3 ⊢ (∃𝑎 ∈ 𝐴 𝑋 = {𝑎} → 𝑋 ∈ V) |
| 9 | eqeq1 2741 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = {𝑎} ↔ 𝑋 = {𝑎})) | |
| 10 | 9 | rexbidv 3179 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑎 ∈ 𝐴 𝑥 = {𝑎} ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎})) |
| 11 | 8, 10 | elab3 3686 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}} ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎}) |
| 12 | 4, 11 | bitrdi 287 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 Vcvv 3480 {csn 4626 ‘cfv 6561 Atomscatm 39264 PointscpointsN 39497 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-ss 3968 df-nul 4334 df-if 4526 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-pointsN 39504 |
| This theorem is referenced by: atpointN 39745 pointpsubN 39753 |
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