![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > pr2nelem | GIF version |
Description: Lemma for pr2ne 7065. (Contributed by FL, 17-Aug-2008.) |
Ref | Expression |
---|---|
pr2nelem | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjsn2 3594 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
2 | ensn1g 6699 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
3 | ensn1g 6699 | . . . . 5 ⊢ (𝐵 ∈ 𝐷 → {𝐵} ≈ 1o) | |
4 | pm54.43 7063 | . . . . . . 7 ⊢ (({𝐴} ≈ 1o ∧ {𝐵} ≈ 1o) → (({𝐴} ∩ {𝐵}) = ∅ ↔ ({𝐴} ∪ {𝐵}) ≈ 2o)) | |
5 | df-pr 3539 | . . . . . . . 8 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
6 | 5 | breq1i 3944 | . . . . . . 7 ⊢ ({𝐴, 𝐵} ≈ 2o ↔ ({𝐴} ∪ {𝐵}) ≈ 2o) |
7 | 4, 6 | syl6bbr 197 | . . . . . 6 ⊢ (({𝐴} ≈ 1o ∧ {𝐵} ≈ 1o) → (({𝐴} ∩ {𝐵}) = ∅ ↔ {𝐴, 𝐵} ≈ 2o)) |
8 | 7 | biimpd 143 | . . . . 5 ⊢ (({𝐴} ≈ 1o ∧ {𝐵} ≈ 1o) → (({𝐴} ∩ {𝐵}) = ∅ → {𝐴, 𝐵} ≈ 2o)) |
9 | 2, 3, 8 | syl2an 287 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (({𝐴} ∩ {𝐵}) = ∅ → {𝐴, 𝐵} ≈ 2o)) |
10 | 9 | ex 114 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (({𝐴} ∩ {𝐵}) = ∅ → {𝐴, 𝐵} ≈ 2o))) |
11 | 1, 10 | syl7 69 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} ≈ 2o))) |
12 | 11 | 3imp 1176 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 ≠ wne 2309 ∪ cun 3074 ∩ cin 3075 ∅c0 3368 {csn 3532 {cpr 3533 class class class wbr 3937 1oc1o 6314 2oc2o 6315 ≈ cen 6640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-1o 6321 df-2o 6322 df-er 6437 df-en 6643 |
This theorem is referenced by: pr2ne 7065 |
Copyright terms: Public domain | W3C validator |