Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > pr2cv | Structured version Visualization version GIF version |
Description: If an unordered pair is equinumerous to ordinal two, then both parts are sets. (Contributed by RP, 8-Oct-2023.) |
Ref | Expression |
---|---|
pr2cv | ⊢ ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2 8751 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o → ∃𝑥∃𝑦{𝐴, 𝐵} = {𝑥, 𝑦}) | |
2 | breq1 5066 | . . . 4 ⊢ ({𝐴, 𝐵} = {𝑥, 𝑦} → ({𝐴, 𝐵} ≈ 2o ↔ {𝑥, 𝑦} ≈ 2o)) | |
3 | vex 3496 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
4 | vex 3496 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | pr2ne 9428 | . . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥, 𝑦} ≈ 2o ↔ 𝑥 ≠ 𝑦)) | |
6 | 5 | el2v 3500 | . . . . . . . 8 ⊢ ({𝑥, 𝑦} ≈ 2o ↔ 𝑥 ≠ 𝑦) |
7 | 6 | biimpi 218 | . . . . . . 7 ⊢ ({𝑥, 𝑦} ≈ 2o → 𝑥 ≠ 𝑦) |
8 | preq12nebg 4790 | . . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑥 ≠ 𝑦) → ({𝑥, 𝑦} = {𝐴, 𝐵} ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴)))) | |
9 | eqvisset 3510 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) | |
10 | eqvisset 3510 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V) | |
11 | 9, 10 | anim12i 614 | . . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
12 | eqvisset 3510 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → 𝐵 ∈ V) | |
13 | eqvisset 3510 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → 𝐴 ∈ V) | |
14 | 12, 13 | anim12ci 615 | . . . . . . . . 9 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
15 | 11, 14 | jaoi 853 | . . . . . . . 8 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
16 | 8, 15 | syl6bi 255 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑥 ≠ 𝑦) → ({𝑥, 𝑦} = {𝐴, 𝐵} → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
17 | 3, 4, 7, 16 | mp3an12i 1460 | . . . . . 6 ⊢ ({𝑥, 𝑦} ≈ 2o → ({𝑥, 𝑦} = {𝐴, 𝐵} → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
18 | 17 | com12 32 | . . . . 5 ⊢ ({𝑥, 𝑦} = {𝐴, 𝐵} → ({𝑥, 𝑦} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
19 | 18 | eqcoms 2828 | . . . 4 ⊢ ({𝐴, 𝐵} = {𝑥, 𝑦} → ({𝑥, 𝑦} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
20 | 2, 19 | sylbid 242 | . . 3 ⊢ ({𝐴, 𝐵} = {𝑥, 𝑦} → ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
21 | 20 | exlimivv 1932 | . 2 ⊢ (∃𝑥∃𝑦{𝐴, 𝐵} = {𝑥, 𝑦} → ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
22 | 1, 21 | mpcom 38 | 1 ⊢ ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1082 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ≠ wne 3015 Vcvv 3493 {cpr 4566 class class class wbr 5063 2oc2o 8093 ≈ cen 8503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3495 df-sbc 3771 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-om 7578 df-1o 8099 df-2o 8100 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 |
This theorem is referenced by: pr2el1 39982 pr2cv1 39983 pr2el2 39984 pr2cv2 39985 pren2 39986 |
Copyright terms: Public domain | W3C validator |