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Mirrors > Home > ILE Home > Th. List > en3i | GIF version |
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.) |
Ref | Expression |
---|---|
en3i.1 | ⊢ 𝐴 ∈ V |
en3i.2 | ⊢ 𝐵 ∈ V |
en3i.3 | ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) |
en3i.4 | ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴) |
en3i.5 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) |
Ref | Expression |
---|---|
en3i | ⊢ 𝐴 ≈ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en3i.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | a1i 9 | . . 3 ⊢ (⊤ → 𝐴 ∈ V) |
3 | en3i.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 3 | a1i 9 | . . 3 ⊢ (⊤ → 𝐵 ∈ V) |
5 | en3i.3 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) | |
6 | 5 | a1i 9 | . . 3 ⊢ (⊤ → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
7 | en3i.4 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴) | |
8 | 7 | a1i 9 | . . 3 ⊢ (⊤ → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴)) |
9 | en3i.5 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) | |
10 | 9 | a1i 9 | . . 3 ⊢ (⊤ → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))) |
11 | 2, 4, 6, 8, 10 | en3d 6796 | . 2 ⊢ (⊤ → 𝐴 ≈ 𝐵) |
12 | 11 | mptru 1373 | 1 ⊢ 𝐴 ≈ 𝐵 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ⊤wtru 1365 ∈ wcel 2160 Vcvv 2752 class class class wbr 4018 ≈ cen 6765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-en 6768 |
This theorem is referenced by: xpmapenlem 6878 nn0ennn 10466 oddennn 12446 evenennn 12447 znnen 12452 |
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