Theorem en3i 8779

Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)

Hypotheses

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))

Assertion

Ref	Expression
en3i	⊢ 𝐴 ≈ 𝐵

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem en3i

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 8777	. 2 ⊢ (⊤ → 𝐴 ≈ 𝐵)
12	11	mptru 1546	1 ⊢ 𝐴 ≈ 𝐵

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ⊤wtru 1540   ∈ wcel 2106  Vcvv 3432   class class class wbr 5074   ≈ cen 8730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-en 8734

This theorem is referenced by:  xpmapenlem  8931  nn0ennn  13699