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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 2824 | . . . 4 ⊢ {𝑋} = {𝑋} | |
2 | sneq 4580 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
3 | 2 | rspceeqv 3641 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ {𝑋} = {𝑋}) → ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥}) |
4 | 1, 3 | mpan2 689 | . . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥}) |
5 | 4 | adantl 484 | . 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 36882 | . . 3 ⊢ (𝐾 ∈ 𝐷 → ({𝑋} ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥})) |
9 | 8 | adantr 483 | . 2 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → ({𝑋} ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 {𝑋} = {𝑥})) |
10 | 5, 9 | mpbird 259 | 1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → {𝑋} ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 {csn 4570 ‘cfv 6358 Atomscatm 36403 PointscpointsN 36635 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-pointsN 36642 |
This theorem is referenced by: (None) |
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