Theorem en2prd 9032

Description: Two proper unordered pairs are equinumerous. (Contributed by BTernaryTau, 23-Dec-2024.)

Hypotheses

Ref	Expression
en2prd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
en2prd.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
en2prd.3	⊢ (𝜑 → 𝐶 ∈ 𝑋)
en2prd.4	⊢ (𝜑 → 𝐷 ∈ 𝑌)
en2prd.5	⊢ (𝜑 → 𝐴 ≠ 𝐵)
en2prd.6	⊢ (𝜑 → 𝐶 ≠ 𝐷)

Assertion

Ref	Expression
en2prd	⊢ (𝜑 → {𝐴, 𝐵} ≈ {𝐶, 𝐷})

Proof of Theorem en2prd

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prex 5400	. . 3 ⊢ {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ∈ V
2		en2prd.5	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
3		en2prd.6	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐷)
4		en2prd.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		en2prd.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
6		en2prd.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		en2prd.4	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
8		f1oprg 6857	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑊 ∧ 𝐷 ∈ 𝑌)) → ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
9	4, 5, 6, 7, 8	syl22anc 851	. . . 4 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
10	2, 3, 9	mp2and 711	. . 3 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
11		f1oeq1 6798	. . . 4 ⊢ (𝑓 = {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} → (𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷} ↔ {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
12	11	spcegv 3559	. . 3 ⊢ ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ∈ V → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷} → ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
13	1, 10, 12	mpsyl 69	. 2 ⊢ (𝜑 → ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
14		prex 5400	. . 3 ⊢ {𝐴, 𝐵} ∈ V
15		prex 5400	. . 3 ⊢ {𝐶, 𝐷} ∈ V
16		breng 8940	. . 3 ⊢ (({𝐴, 𝐵} ∈ V ∧ {𝐶, 𝐷} ∈ V) → ({𝐴, 𝐵} ≈ {𝐶, 𝐷} ↔ ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
17	14, 15, 16	mp2an 704	. 2 ⊢ ({𝐴, 𝐵} ≈ {𝐶, 𝐷} ↔ ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
18	13, 17	sylibr 237	1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ {𝐶, 𝐷})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∃wex 1802   ∈ wcel 2145   ≠ wne 2960  Vcvv 3457  {cpr 4587  ⟨cop 4591   class class class wbr 5105  –1-1-onto→wf1o 6524   ≈ cen 8928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-en 8932

This theorem is referenced by:  enpr2d  9033  rex2dom  9201