Theorem en2prd 6968

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 4313	. . . . 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 4313	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐷 ∈ 𝑌) → ⟨𝐵, 𝐷⟩ ∈ V)
8	5, 6, 7	syl2anc 411	. . . 4 ⊢ (𝜑 → ⟨𝐵, 𝐷⟩ ∈ V)
9		prexg 4294	. . . 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 5616	. . . . 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 1272	. . . 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 5559	. . 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 2948	. 2 ⊢ (𝜑 → ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
18		prexg 4294	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
19	1, 5, 18	syl2anc 411	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ V)
20		prexg 4294	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → {𝐶, 𝐷} ∈ V)
21	2, 6, 20	syl2anc 411	. . 3 ⊢ (𝜑 → {𝐶, 𝐷} ∈ V)
22		breng 6892	. . 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 1538   ∈ wcel 2200   ≠ wne 2400  Vcvv 2799  {cpr 3667  ⟨cop 3669   class class class wbr 4082  –1-1-onto→wf1o 5316   ≈ cen 6883

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-en 6886

This theorem is referenced by:  rex2dom  6969