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Mirrors > Home > MPE Home > Th. List > hashle2pr | Structured version Visualization version GIF version |
Description: A nonempty set of size less than or equal to two is an unordered pair of sets. (Contributed by AV, 24-Nov-2021.) |
Ref | Expression |
---|---|
hashle2pr | ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑃 ≠ ∅) → ((♯‘𝑃) ≤ 2 ↔ ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashxnn0 13695 | . . . . . . 7 ⊢ (𝑃 ∈ 𝑉 → (♯‘𝑃) ∈ ℕ0*) | |
2 | xnn0le2is012 12627 | . . . . . . 7 ⊢ (((♯‘𝑃) ∈ ℕ0* ∧ (♯‘𝑃) ≤ 2) → ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ (♯‘𝑃) = 2)) | |
3 | 1, 2 | sylan 583 | . . . . . 6 ⊢ ((𝑃 ∈ 𝑉 ∧ (♯‘𝑃) ≤ 2) → ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ (♯‘𝑃) = 2)) |
4 | 3 | ex 416 | . . . . 5 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) ≤ 2 → ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ (♯‘𝑃) = 2))) |
5 | hasheq0 13720 | . . . . . . . . 9 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) = 0 ↔ 𝑃 = ∅)) | |
6 | eqneqall 2998 | . . . . . . . . 9 ⊢ (𝑃 = ∅ → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏})) | |
7 | 5, 6 | syl6bi 256 | . . . . . . . 8 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) = 0 → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}))) |
8 | 7 | com12 32 | . . . . . . 7 ⊢ ((♯‘𝑃) = 0 → (𝑃 ∈ 𝑉 → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}))) |
9 | hash1snb 13776 | . . . . . . . . . . 11 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) = 1 ↔ ∃𝑐 𝑃 = {𝑐})) | |
10 | vex 3444 | . . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V | |
11 | preq12 4631 | . . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑐) → {𝑎, 𝑏} = {𝑐, 𝑐}) | |
12 | dfsn2 4538 | . . . . . . . . . . . . . . 15 ⊢ {𝑐} = {𝑐, 𝑐} | |
13 | 11, 12 | eqtr4di 2851 | . . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑐) → {𝑎, 𝑏} = {𝑐}) |
14 | 13 | eqeq2d 2809 | . . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑐) → (𝑃 = {𝑎, 𝑏} ↔ 𝑃 = {𝑐})) |
15 | 10, 10, 14 | spc2ev 3556 | . . . . . . . . . . . 12 ⊢ (𝑃 = {𝑐} → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}) |
16 | 15 | exlimiv 1931 | . . . . . . . . . . 11 ⊢ (∃𝑐 𝑃 = {𝑐} → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}) |
17 | 9, 16 | syl6bi 256 | . . . . . . . . . 10 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) = 1 → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏})) |
18 | 17 | imp 410 | . . . . . . . . 9 ⊢ ((𝑃 ∈ 𝑉 ∧ (♯‘𝑃) = 1) → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}) |
19 | 18 | a1d 25 | . . . . . . . 8 ⊢ ((𝑃 ∈ 𝑉 ∧ (♯‘𝑃) = 1) → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏})) |
20 | 19 | expcom 417 | . . . . . . 7 ⊢ ((♯‘𝑃) = 1 → (𝑃 ∈ 𝑉 → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}))) |
21 | hash2pr 13823 | . . . . . . . . 9 ⊢ ((𝑃 ∈ 𝑉 ∧ (♯‘𝑃) = 2) → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}) | |
22 | 21 | a1d 25 | . . . . . . . 8 ⊢ ((𝑃 ∈ 𝑉 ∧ (♯‘𝑃) = 2) → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏})) |
23 | 22 | expcom 417 | . . . . . . 7 ⊢ ((♯‘𝑃) = 2 → (𝑃 ∈ 𝑉 → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}))) |
24 | 8, 20, 23 | 3jaoi 1424 | . . . . . 6 ⊢ (((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ (♯‘𝑃) = 2) → (𝑃 ∈ 𝑉 → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}))) |
25 | 24 | com12 32 | . . . . 5 ⊢ (𝑃 ∈ 𝑉 → (((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ (♯‘𝑃) = 2) → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}))) |
26 | 4, 25 | syld 47 | . . . 4 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) ≤ 2 → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}))) |
27 | 26 | com23 86 | . . 3 ⊢ (𝑃 ∈ 𝑉 → (𝑃 ≠ ∅ → ((♯‘𝑃) ≤ 2 → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}))) |
28 | 27 | imp 410 | . 2 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑃 ≠ ∅) → ((♯‘𝑃) ≤ 2 → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏})) |
29 | fveq2 6645 | . . . 4 ⊢ (𝑃 = {𝑎, 𝑏} → (♯‘𝑃) = (♯‘{𝑎, 𝑏})) | |
30 | hashprlei 13822 | . . . . 5 ⊢ ({𝑎, 𝑏} ∈ Fin ∧ (♯‘{𝑎, 𝑏}) ≤ 2) | |
31 | 30 | simpri 489 | . . . 4 ⊢ (♯‘{𝑎, 𝑏}) ≤ 2 |
32 | 29, 31 | eqbrtrdi 5069 | . . 3 ⊢ (𝑃 = {𝑎, 𝑏} → (♯‘𝑃) ≤ 2) |
33 | 32 | exlimivv 1933 | . 2 ⊢ (∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏} → (♯‘𝑃) ≤ 2) |
34 | 28, 33 | impbid1 228 | 1 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑃 ≠ ∅) → ((♯‘𝑃) ≤ 2 ↔ ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ w3o 1083 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 {csn 4525 {cpr 4527 class class class wbr 5030 ‘cfv 6324 Fincfn 8492 0cc0 10526 1c1 10527 ≤ cle 10665 2c2 11680 ℕ0*cxnn0 11955 ♯chash 13686 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 |
This theorem is referenced by: hashle2prv 13832 |
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