Theorem en3i 6542

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 6540	. 2 ⊢ (⊤ → 𝐴 ≈ 𝐵)
12	11	mptru 1299	1 ⊢ 𝐴 ≈ 𝐵

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1290  ⊤wtru 1291   ∈ wcel 1439  Vcvv 2620   class class class wbr 3851   ≈ cen 6509

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-en 6512

This theorem is referenced by:  xpmapenlem  6619  nn0ennn  9894  oddennn  11537  evenennn  11538  znnen  11543