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Mirrors > Home > MPE Home > Th. List > en3i | Structured version Visualization version 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 11 | . . 3 ⊢ (⊤ → 𝐴 ∈ V) |
3 | en3i.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → 𝐵 ∈ V) |
5 | en3i.3 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
7 | en3i.4 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴) | |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴)) |
9 | en3i.5 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) | |
10 | 9 | a1i 11 | . . 3 ⊢ (⊤ → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))) |
11 | 2, 4, 6, 8, 10 | en3d 8981 | . 2 ⊢ (⊤ → 𝐴 ≈ 𝐵) |
12 | 11 | mptru 1540 | 1 ⊢ 𝐴 ≈ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 Vcvv 3466 class class class wbr 5138 ≈ cen 8932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-en 8936 |
This theorem is referenced by: xpmapenlem 9140 nn0ennn 13941 |
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