Theorem en2prd 8985

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 5373	. . 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 6818	. . . . 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 839	. . . 4 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
10	2, 3, 9	mp2and 700	. . 3 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
11		f1oeq1 6760	. . . 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 3540	. . 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 68	. 2 ⊢ (𝜑 → ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
14		prex 5373	. . 3 ⊢ {𝐴, 𝐵} ∈ V
15		prex 5373	. . 3 ⊢ {𝐶, 𝐷} ∈ V
16		breng 8893	. . 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 693	. 2 ⊢ ({𝐴, 𝐵} ≈ {𝐶, 𝐷} ↔ ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
18	13, 17	sylibr 234	1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ {𝐶, 𝐷})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  Vcvv 3430  {cpr 4570  ⟨cop 4574   class class class wbr 5086  –1-1-onto→wf1o 6489   ≈ cen 8881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-en 8885

This theorem is referenced by:  enpr2d  8986  rex2dom  9154