Theorem en2prd 7096

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		en2prd.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		en2prd.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
3		opexg 4363	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) → ⟨𝐴, 𝐶⟩ ∈ V)
4	1, 2, 3	syl2anc 415	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ ∈ V)
5		en2prd.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6		en2prd.4	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
7		opexg 4363	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐷 ∈ 𝑌) → ⟨𝐵, 𝐷⟩ ∈ V)
8	5, 6, 7	syl2anc 415	. . . 4 ⊢ (𝜑 → ⟨𝐵, 𝐷⟩ ∈ V)
9		prexg 4344	. . . 4 ⊢ ((⟨𝐴, 𝐶⟩ ∈ V ∧ ⟨𝐵, 𝐷⟩ ∈ V) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ∈ V)
10	4, 8, 9	syl2anc 415	. . 3 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ∈ V)
11		en2prd.5	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
12		en2prd.6	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐷)
13		f1oprg 5680	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑊 ∧ 𝐷 ∈ 𝑌)) → ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
14	1, 2, 5, 6, 13	syl22anc 1279	. . . 4 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
15	11, 12, 14	mp2and 437	. . 3 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
16		f1oeq1 5622	. . 3 ⊢ (𝑓 = {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} → (𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷} ↔ {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
17	10, 15, 16	elabd 2971	. 2 ⊢ (𝜑 → ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
18		prexg 4344	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
19	1, 5, 18	syl2anc 415	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ V)
20		prexg 4344	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → {𝐶, 𝐷} ∈ V)
21	2, 6, 20	syl2anc 415	. . 3 ⊢ (𝜑 → {𝐶, 𝐷} ∈ V)
22		breng 7019	. . 3 ⊢ (({𝐴, 𝐵} ∈ V ∧ {𝐶, 𝐷} ∈ V) → ({𝐴, 𝐵} ≈ {𝐶, 𝐷} ↔ ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
23	19, 21, 22	syl2anc 415	. 2 ⊢ (𝜑 → ({𝐴, 𝐵} ≈ {𝐶, 𝐷} ↔ ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
24	17, 23	mpbird 167	1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ {𝐶, 𝐷})

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1545   ∈ wcel 2209   ≠ wne 2420  Vcvv 2821  {cpr 3706  ⟨cop 3708   class class class wbr 4125  –1-1-onto→wf1o 5371   ≈ cen 7010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  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 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-en 7013

This theorem is referenced by:  rex2dom  7100