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Mirrors > Home > MPE Home > Th. List > en2i | Structured version Visualization version GIF version |
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.) |
Ref | Expression |
---|---|
en2i.1 | ⊢ 𝐴 ∈ V |
en2i.2 | ⊢ 𝐵 ∈ V |
en2i.3 | ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ V) |
en2i.4 | ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ V) |
en2i.5 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) |
Ref | Expression |
---|---|
en2i | ⊢ 𝐴 ≈ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2i.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → 𝐴 ∈ V) |
3 | en2i.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → 𝐵 ∈ V) |
5 | en2i.3 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ V) | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → (𝑥 ∈ 𝐴 → 𝐶 ∈ V)) |
7 | en2i.4 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ V) | |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → (𝑦 ∈ 𝐵 → 𝐷 ∈ V)) |
9 | en2i.5 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) | |
10 | 9 | a1i 11 | . . 3 ⊢ (⊤ → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) |
11 | 2, 4, 6, 8, 10 | en2d 7877 | . 2 ⊢ (⊤ → 𝐴 ≈ 𝐵) |
12 | 11 | trud 1484 | 1 ⊢ 𝐴 ≈ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ≈ cen 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-en 7842 |
This theorem is referenced by: mapsnen 7920 xpsnen 7929 xpassen 7939 |
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