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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 34045 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑃 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}}) |
4 | 3 | eleq2d 2673 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}})) |
5 | snex 4835 | . . . . 5 ⊢ {𝑎} ∈ V | |
6 | eleq1 2676 | . . . . 5 ⊢ (𝑋 = {𝑎} → (𝑋 ∈ V ↔ {𝑎} ∈ V)) | |
7 | 5, 6 | mpbiri 247 | . . . 4 ⊢ (𝑋 = {𝑎} → 𝑋 ∈ V) |
8 | 7 | rexlimivw 3011 | . . 3 ⊢ (∃𝑎 ∈ 𝐴 𝑋 = {𝑎} → 𝑋 ∈ V) |
9 | eqeq1 2614 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = {𝑎} ↔ 𝑋 = {𝑎})) | |
10 | 9 | rexbidv 3034 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑎 ∈ 𝐴 𝑥 = {𝑎} ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎})) |
11 | 8, 10 | elab3 3327 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}} ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎}) |
12 | 4, 11 | syl6bb 275 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 {cab 2596 ∃wrex 2897 Vcvv 3173 {csn 4125 ‘cfv 5804 Atomscatm 33568 PointscpointsN 33799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-pointsN 33806 |
This theorem is referenced by: atpointN 34047 pointpsubN 34055 |
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