Theorem en3i 7023

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 9	. . 3 ⊢ (⊤ → 𝐴 ∈ V)
3		en3i.2	. . . 4 ⊢ 𝐵 ∈ V
4	3	a1i 9	. . 3 ⊢ (⊤ → 𝐵 ∈ V)
5		en3i.3	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)
6	5	a1i 9	. . 3 ⊢ (⊤ → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))
7		en3i.4	. . . 4 ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴)
8	7	a1i 9	. . 3 ⊢ (⊤ → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴))
9		en3i.5	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))
10	9	a1i 9	. . 3 ⊢ (⊤ → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)))
11	2, 4, 6, 8, 10	en3d 7021	. 2 ⊢ (⊤ → 𝐴 ≈ 𝐵)
12	11	mptru 1407	1 ⊢ 𝐴 ≈ 𝐵

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ⊤wtru 1399   ∈ wcel 2205  Vcvv 2815   class class class wbr 4114   ≈ cen 6986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-en 6989

This theorem is referenced by:  xpmapenlem  7115  nn0ennn  10819  oddennn  13227  evenennn  13228  znnen  13233