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Mirrors > Home > MPE Home > Th. List > hash1elsn | Structured version Visualization version GIF version |
Description: A set of size 1 with a known element is the singleton of that element. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
hash1elsn.1 | ⊢ (𝜑 → (♯‘𝐴) = 1) |
hash1elsn.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
hash1elsn.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
hash1elsn | ⊢ (𝜑 → 𝐴 = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hash1elsn.1 | . . . 4 ⊢ (𝜑 → (♯‘𝐴) = 1) | |
2 | hash1elsn.3 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | hashen1 14270 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o)) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o)) |
5 | 1, 4 | mpbid 231 | . . 3 ⊢ (𝜑 → 𝐴 ≈ 1o) |
6 | en1 8965 | . . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | |
7 | 5, 6 | sylib 217 | . 2 ⊢ (𝜑 → ∃𝑥 𝐴 = {𝑥}) |
8 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = {𝑥}) → 𝐴 = {𝑥}) | |
9 | hash1elsn.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
10 | 9 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = {𝑥}) → 𝐵 ∈ 𝐴) |
11 | 10, 8 | eleqtrd 2840 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = {𝑥}) → 𝐵 ∈ {𝑥}) |
12 | elsni 4603 | . . . . 5 ⊢ (𝐵 ∈ {𝑥} → 𝐵 = 𝑥) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = {𝑥}) → 𝐵 = 𝑥) |
14 | 13 | sneqd 4598 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = {𝑥}) → {𝐵} = {𝑥}) |
15 | 8, 14 | eqtr4d 2779 | . 2 ⊢ ((𝜑 ∧ 𝐴 = {𝑥}) → 𝐴 = {𝐵}) |
16 | 7, 15 | exlimddv 1938 | 1 ⊢ (𝜑 → 𝐴 = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {csn 4586 class class class wbr 5105 ‘cfv 6496 1oc1o 8405 ≈ cen 8880 1c1 11052 ♯chash 14230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-hash 14231 |
This theorem is referenced by: prmgrpsimpgd 19893 ablsimpgprmd 19894 |
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