Theorem en2i 8921

Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)

Hypotheses

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))

Assertion

Ref	Expression
en2i	⊢ 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 en2i

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 8919	. 2 ⊢ (⊤ → 𝐴 ≈ 𝐵)
12	11	mptru 1548	1 ⊢ 𝐴 ≈ 𝐵

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ⊤wtru 1542   ∈ wcel 2113  Vcvv 3437   class class class wbr 5095   ≈ cen 8874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-en 8878

This theorem is referenced by:  xpsnen  8983  xpassen  8993