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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 37441 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑃 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}}) |
4 | 3 | eleq2d 2816 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}})) |
5 | snex 5309 | . . . . 5 ⊢ {𝑎} ∈ V | |
6 | eleq1 2818 | . . . . 5 ⊢ (𝑋 = {𝑎} → (𝑋 ∈ V ↔ {𝑎} ∈ V)) | |
7 | 5, 6 | mpbiri 261 | . . . 4 ⊢ (𝑋 = {𝑎} → 𝑋 ∈ V) |
8 | 7 | rexlimivw 3191 | . . 3 ⊢ (∃𝑎 ∈ 𝐴 𝑋 = {𝑎} → 𝑋 ∈ V) |
9 | eqeq1 2740 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = {𝑎} ↔ 𝑋 = {𝑎})) | |
10 | 9 | rexbidv 3206 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑎 ∈ 𝐴 𝑥 = {𝑎} ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎})) |
11 | 8, 10 | elab3 3584 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}} ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎}) |
12 | 4, 11 | bitrdi 290 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 {cab 2714 ∃wrex 3052 Vcvv 3398 {csn 4527 ‘cfv 6358 Atomscatm 36963 PointscpointsN 37195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-pointsN 37202 |
This theorem is referenced by: atpointN 37443 pointpsubN 37451 |
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