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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 6747 | . 2 ⊢ (⊤ → 𝐴 ≈ 𝐵) |
12 | 11 | mptru 1357 | 1 ⊢ 𝐴 ≈ 𝐵 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ⊤wtru 1349 ∈ wcel 2141 Vcvv 2730 class class class wbr 3989 ≈ cen 6716 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-en 6719 |
This theorem is referenced by: xpmapenlem 6827 nn0ennn 10389 oddennn 12347 evenennn 12348 znnen 12353 |
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