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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 13831 | . . 3 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) = 2 ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑥 ≠ 𝑦 ∧ 𝑃 = {𝑥, 𝑦}))) | |
2 | simpr 487 | . . . . . . . 8 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑃 = {𝑥, 𝑦}) → 𝑃 = {𝑥, 𝑦}) | |
3 | 3simpa 1144 | . . . . . . . . . . 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 2901 | . . . . . . . . . . . 12 ⊢ (𝑃 = {𝑥, 𝑦} → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {𝑥, 𝑦})) | |
6 | eleq2 2901 | . . . . . . . . . . . 12 ⊢ (𝑃 = {𝑥, 𝑦} → (𝑌 ∈ 𝑃 ↔ 𝑌 ∈ {𝑥, 𝑦})) | |
7 | 5, 6 | anbi12d 632 | . . . . . . . . . . 11 ⊢ (𝑃 = {𝑥, 𝑦} → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ↔ (𝑋 ∈ {𝑥, 𝑦} ∧ 𝑌 ∈ {𝑥, 𝑦}))) |
8 | 7 | adantl 484 | . . . . . . . . . 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 234 | . . . . . . . . 9 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑃 = {𝑥, 𝑦}) → (𝑋 ∈ {𝑥, 𝑦} ∧ 𝑌 ∈ {𝑥, 𝑦})) |
10 | prel12g 4794 | . . . . . . . . . 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 259 | . . . . . . . 8 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑃 = {𝑥, 𝑦}) → {𝑋, 𝑌} = {𝑥, 𝑦}) |
13 | 2, 12 | eqtr4d 2859 | . . . . . . 7 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑃 = {𝑥, 𝑦}) → 𝑃 = {𝑋, 𝑌}) |
14 | 13 | exp31 422 | . . . . . 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 456 | . . . 4 ⊢ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) → ((𝑥 ≠ 𝑦 ∧ 𝑃 = {𝑥, 𝑦}) → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → 𝑃 = {𝑋, 𝑌}))) |
17 | 16 | rexlimivv 3292 | . . 3 ⊢ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑥 ≠ 𝑦 ∧ 𝑃 = {𝑥, 𝑦}) → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → 𝑃 = {𝑋, 𝑌})) |
18 | 1, 17 | syl6bi 255 | . 2 ⊢ (𝑃 ∈ 𝑉 → ((♯‘𝑃) = 2 → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → 𝑃 = {𝑋, 𝑌}))) |
19 | 18 | imp 409 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 {cpr 4569 ‘cfv 6355 2c2 11693 ♯chash 13691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 |
This theorem is referenced by: symg2bas 18521 |
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