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Mirrors > Home > MPE Home > Th. List > en2prd | Structured version Visualization version GIF version |
Description: Two unordered pairs are equinumerous. (Contributed by BTernaryTau, 23-Dec-2024.) |
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) |
Ref | Expression |
---|---|
en2prd | ⊢ (𝜑 → {𝐴, 𝐵} ≈ {𝐶, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prex 5425 | . . 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 6865 | . . . . 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 837 | . . . 4 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})) |
10 | 2, 3, 9 | mp2and 697 | . . 3 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}) |
11 | f1oeq1 6808 | . . . 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 3584 | . . 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 5425 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
15 | prex 5425 | . . 3 ⊢ {𝐶, 𝐷} ∈ V | |
16 | breng 8931 | . . 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 690 | . 2 ⊢ ({𝐴, 𝐵} ≈ {𝐶, 𝐷} ↔ ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}) |
18 | 13, 17 | sylibr 233 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ {𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∃wex 1781 ∈ wcel 2106 ≠ wne 2939 Vcvv 3473 {cpr 4624 ⟨cop 4628 class class class wbr 5141 –1-1-onto→wf1o 6531 ≈ cen 8919 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-en 8923 |
This theorem is referenced by: enpr2d 9032 rex2dom 9229 |
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