Theorem en3i 9002

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 9000	. 2 ⊢ (⊤ → 𝐴 ≈ 𝐵)
12	11	mptru 1577	1 ⊢ 𝐴 ≈ 𝐵

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145  Vcvv 3451   class class class wbr 5103   ≈ cen 8954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-en 8958

This theorem is used by:  xpmapenlem  9147  nn0ennn  14102