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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 14278 | . . . . . . 7 ⊢ (𝑃 ∈ 𝑉 → (♯‘𝑃) ∈ ℕ0*) | |
2 | xnn0le2is012 13204 | . . . . . . 7 ⊢ (((♯‘𝑃) ∈ ℕ0* ∧ (♯‘𝑃) ≤ 2) → ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ (♯‘𝑃) = 2)) | |
3 | 1, 2 | sylan 580 | . . . . . 6 ⊢ ((𝑃 ∈ 𝑉 ∧ (♯‘𝑃) ≤ 2) → ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ (♯‘𝑃) = 2)) |
4 | 3 | ex 413 | . . . . 5 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) ≤ 2 → ((♯‘𝑃) = 0 ∨ (♯‘𝑃) = 1 ∨ (♯‘𝑃) = 2))) |
5 | hasheq0 14302 | . . . . . . . . 9 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) = 0 ↔ 𝑃 = ∅)) | |
6 | eqneqall 2950 | . . . . . . . . 9 ⊢ (𝑃 = ∅ → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏})) | |
7 | 5, 6 | syl6bi 252 | . . . . . . . 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 14358 | . . . . . . . . . . 11 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) = 1 ↔ ∃𝑐 𝑃 = {𝑐})) | |
10 | vex 3474 | . . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V | |
11 | preq12 4729 | . . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑐) → {𝑎, 𝑏} = {𝑐, 𝑐}) | |
12 | dfsn2 4632 | . . . . . . . . . . . . . . 15 ⊢ {𝑐} = {𝑐, 𝑐} | |
13 | 11, 12 | eqtr4di 2789 | . . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑐) → {𝑎, 𝑏} = {𝑐}) |
14 | 13 | eqeq2d 2742 | . . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑐) → (𝑃 = {𝑎, 𝑏} ↔ 𝑃 = {𝑐})) |
15 | 10, 10, 14 | spc2ev 3591 | . . . . . . . . . . . 12 ⊢ (𝑃 = {𝑐} → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}) |
16 | 15 | exlimiv 1933 | . . . . . . . . . . 11 ⊢ (∃𝑐 𝑃 = {𝑐} → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}) |
17 | 9, 16 | syl6bi 252 | . . . . . . . . . 10 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) = 1 → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏})) |
18 | 17 | imp 407 | . . . . . . . . 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 414 | . . . . . . 7 ⊢ ((♯‘𝑃) = 1 → (𝑃 ∈ 𝑉 → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}))) |
21 | hash2pr 14409 | . . . . . . . . 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 414 | . . . . . . 7 ⊢ ((♯‘𝑃) = 2 → (𝑃 ∈ 𝑉 → (𝑃 ≠ ∅ → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}))) |
24 | 8, 20, 23 | 3jaoi 1427 | . . . . . 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 407 | . 2 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑃 ≠ ∅) → ((♯‘𝑃) ≤ 2 → ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏})) |
29 | fveq2 6875 | . . . 4 ⊢ (𝑃 = {𝑎, 𝑏} → (♯‘𝑃) = (♯‘{𝑎, 𝑏})) | |
30 | hashprlei 14408 | . . . . 5 ⊢ ({𝑎, 𝑏} ∈ Fin ∧ (♯‘{𝑎, 𝑏}) ≤ 2) | |
31 | 30 | simpri 486 | . . . 4 ⊢ (♯‘{𝑎, 𝑏}) ≤ 2 |
32 | 29, 31 | eqbrtrdi 5177 | . . 3 ⊢ (𝑃 = {𝑎, 𝑏} → (♯‘𝑃) ≤ 2) |
33 | 32 | exlimivv 1935 | . 2 ⊢ (∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏} → (♯‘𝑃) ≤ 2) |
34 | 28, 33 | impbid1 224 | 1 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑃 ≠ ∅) → ((♯‘𝑃) ≤ 2 ↔ ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ w3o 1086 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2939 ∅c0 4315 {csn 4619 {cpr 4621 class class class wbr 5138 ‘cfv 6529 Fincfn 8919 0cc0 11089 1c1 11090 ≤ cle 11228 2c2 12246 ℕ0*cxnn0 12523 ♯chash 14269 |
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 2702 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7836 df-1st 7954 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-1o 8445 df-2o 8446 df-oadd 8449 df-er 8683 df-en 8920 df-dom 8921 df-sdom 8922 df-fin 8923 df-dju 9875 df-card 9913 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-nn 12192 df-2 12254 df-n0 12452 df-xnn0 12524 df-z 12538 df-uz 12802 df-fz 13464 df-hash 14270 |
This theorem is referenced by: hashle2prv 14418 |
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