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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 7991 | . 2 ⊢ (⊤ → 𝐴 ≈ 𝐵) |
12 | 11 | trud 1493 | 1 ⊢ 𝐴 ≈ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 ≈ cen 7952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-en 7956 |
This theorem is referenced by: mapsnen 8035 xpsnen 8044 xpassen 8054 |
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