Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 8539 | . 2 ⊢ (⊤ → 𝐴 ≈ 𝐵) |
12 | 11 | mptru 1540 | 1 ⊢ 𝐴 ≈ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 ≈ cen 8500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-en 8504 |
This theorem is referenced by: xpsnen 8595 xpassen 8605 |
Copyright terms: Public domain | W3C validator |