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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 6540 | . 2 ⊢ (⊤ → 𝐴 ≈ 𝐵) |
12 | 11 | mptru 1299 | 1 ⊢ 𝐴 ≈ 𝐵 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1290 ⊤wtru 1291 ∈ wcel 1439 Vcvv 2620 class class class wbr 3851 ≈ cen 6509 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-en 6512 |
This theorem is referenced by: xpmapenlem 6619 nn0ennn 9894 oddennn 11537 evenennn 11538 znnen 11543 |
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