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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 8259 | . 2 ⊢ (⊤ → 𝐴 ≈ 𝐵) |
12 | 11 | mptru 1666 | 1 ⊢ 𝐴 ≈ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ⊤wtru 1659 ∈ wcel 2166 Vcvv 3414 class class class wbr 4873 ≈ cen 8219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-en 8223 |
This theorem is referenced by: xpmapenlem 8396 nn0ennn 13073 |
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