en3i - Intuitionistic Logic Explorer

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Theorem en3i 7057

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 7055	. 2 ⊢ (⊤ → 𝐴 ≈ 𝐵)
12	11	mptru 1411	1 ⊢ 𝐴 ≈ 𝐵

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ⊤wtru 1403 ∈ wcel 2209 Vcvv 2821 class class class wbr 4130 ≈ cen 7020

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-v 2823 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-en 7023

This theorem is used by: xpmapenlem 7149 nn0ennn 10870 oddennn 13283 evenennn 13284 znnen 13289

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