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Mirrors > Home > MPE Home > Th. List > Mathboxes > atpointN | Structured version Visualization version GIF version |
Description: The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ispoint.a | ⊢ 𝐴 = (Atoms‘𝐾) |
ispoint.p | ⊢ 𝑃 = (Points‘𝐾) |
Ref | Expression |
---|---|
atpointN | ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → {𝑋} ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . 4 ⊢ {𝑋} = {𝑋} | |
2 | sneq 4526 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
3 | 2 | rspceeqv 3541 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ {𝑋} = {𝑋}) → ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥}) |
4 | 1, 3 | mpan2 691 | . . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥}) |
5 | 4 | adantl 485 | . 2 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥}) |
6 | ispoint.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | ispoint.p | . . . 4 ⊢ 𝑃 = (Points‘𝐾) | |
8 | 6, 7 | ispointN 37379 | . . 3 ⊢ (𝐾 ∈ 𝐷 → ({𝑋} ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥})) |
9 | 8 | adantr 484 | . 2 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → ({𝑋} ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥})) |
10 | 5, 9 | mpbird 260 | 1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → {𝑋} ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∃wrex 3054 {csn 4516 ‘cfv 6339 Atomscatm 36900 PointscpointsN 37132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-pointsN 37139 |
This theorem is referenced by: (None) |
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