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Mirrors > Home > MPE Home > Th. List > hash2prd | Structured version Visualization version GIF version |
Description: A set of size two is an unordered pair if it contains two different elements. (Contributed by Alexander van der Vekens, 9-Dec-2018.) (Proof shortened by AV, 16-Jun-2022.) |
Ref | Expression |
---|---|
hash2prd | ⊢ ((𝑃 ∈ 𝑉 ∧ (♯‘𝑃) = 2) → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → 𝑃 = {𝑋, 𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hash2prb 14371 | . . 3 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) = 2 ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑥 ≠ 𝑦 ∧ 𝑃 = {𝑥, 𝑦}))) | |
2 | simpr 485 | . . . . . . . 8 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑃 = {𝑥, 𝑦}) → 𝑃 = {𝑥, 𝑦}) | |
3 | 3simpa 1148 | . . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃)) | |
4 | 3 | ad2antlr 725 | . . . . . . . . . 10 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑃 = {𝑥, 𝑦}) → (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃)) |
5 | eleq2 2826 | . . . . . . . . . . . 12 ⊢ (𝑃 = {𝑥, 𝑦} → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {𝑥, 𝑦})) | |
6 | eleq2 2826 | . . . . . . . . . . . 12 ⊢ (𝑃 = {𝑥, 𝑦} → (𝑌 ∈ 𝑃 ↔ 𝑌 ∈ {𝑥, 𝑦})) | |
7 | 5, 6 | anbi12d 631 | . . . . . . . . . . 11 ⊢ (𝑃 = {𝑥, 𝑦} → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ↔ (𝑋 ∈ {𝑥, 𝑦} ∧ 𝑌 ∈ {𝑥, 𝑦}))) |
8 | 7 | adantl 482 | . . . . . . . . . 10 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑃 = {𝑥, 𝑦}) → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ↔ (𝑋 ∈ {𝑥, 𝑦} ∧ 𝑌 ∈ {𝑥, 𝑦}))) |
9 | 4, 8 | mpbid 231 | . . . . . . . . 9 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑃 = {𝑥, 𝑦}) → (𝑋 ∈ {𝑥, 𝑦} ∧ 𝑌 ∈ {𝑥, 𝑦})) |
10 | prel12g 4821 | . . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ({𝑋, 𝑌} = {𝑥, 𝑦} ↔ (𝑋 ∈ {𝑥, 𝑦} ∧ 𝑌 ∈ {𝑥, 𝑦}))) | |
11 | 10 | ad2antlr 725 | . . . . . . . . 9 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑃 = {𝑥, 𝑦}) → ({𝑋, 𝑌} = {𝑥, 𝑦} ↔ (𝑋 ∈ {𝑥, 𝑦} ∧ 𝑌 ∈ {𝑥, 𝑦}))) |
12 | 9, 11 | mpbird 256 | . . . . . . . 8 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑃 = {𝑥, 𝑦}) → {𝑋, 𝑌} = {𝑥, 𝑦}) |
13 | 2, 12 | eqtr4d 2779 | . . . . . . 7 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑃 = {𝑥, 𝑦}) → 𝑃 = {𝑋, 𝑌}) |
14 | 13 | exp31 420 | . . . . . 6 ⊢ (((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑃 = {𝑥, 𝑦} → 𝑃 = {𝑋, 𝑌}))) |
15 | 14 | com23 86 | . . . . 5 ⊢ (((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → (𝑃 = {𝑥, 𝑦} → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → 𝑃 = {𝑋, 𝑌}))) |
16 | 15 | expimpd 454 | . . . 4 ⊢ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) → ((𝑥 ≠ 𝑦 ∧ 𝑃 = {𝑥, 𝑦}) → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → 𝑃 = {𝑋, 𝑌}))) |
17 | 16 | rexlimivv 3196 | . . 3 ⊢ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑥 ≠ 𝑦 ∧ 𝑃 = {𝑥, 𝑦}) → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → 𝑃 = {𝑋, 𝑌})) |
18 | 1, 17 | syl6bi 252 | . 2 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) = 2 → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → 𝑃 = {𝑋, 𝑌}))) |
19 | 18 | imp 407 | 1 ⊢ ((𝑃 ∈ 𝑉 ∧ (♯‘𝑃) = 2) → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → 𝑃 = {𝑋, 𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 {cpr 4588 ‘cfv 6496 2c2 12208 ♯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-2o 8413 df-oadd 8416 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 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-2 12216 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-hash 14231 |
This theorem is referenced by: symg2bas 19174 |
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