Theorem en2prd 7072

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 4349	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) → ⟨𝐴, 𝐶⟩ ∈ V)
4	1, 2, 3	syl2anc 411	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ ∈ V)
5		en2prd.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6		en2prd.4	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
7		opexg 4349	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐷 ∈ 𝑌) → ⟨𝐵, 𝐷⟩ ∈ V)
8	5, 6, 7	syl2anc 411	. . . 4 ⊢ (𝜑 → ⟨𝐵, 𝐷⟩ ∈ V)
9		prexg 4330	. . . 4 ⊢ ((⟨𝐴, 𝐶⟩ ∈ V ∧ ⟨𝐵, 𝐷⟩ ∈ V) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ∈ V)
10	4, 8, 9	syl2anc 411	. . 3 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ∈ V)
11		en2prd.5	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
12		en2prd.6	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐷)
13		f1oprg 5665	. . . . 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 1275	. . . 4 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
15	11, 12, 14	mp2and 433	. . 3 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
16		f1oeq1 5607	. . 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 2965	. 2 ⊢ (𝜑 → ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
18		prexg 4330	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
19	1, 5, 18	syl2anc 411	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ V)
20		prexg 4330	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → {𝐶, 𝐷} ∈ V)
21	2, 6, 20	syl2anc 411	. . 3 ⊢ (𝜑 → {𝐶, 𝐷} ∈ V)
22		breng 6995	. . 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 411	. 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 1541   ∈ wcel 2205   ≠ wne 2414  Vcvv 2815  {cpr 3695  ⟨cop 3697   class class class wbr 4114  –1-1-onto→wf1o 5356   ≈ cen 6986

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-en 6989

This theorem is referenced by:  rex2dom  7076