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Mirrors > Home > MPE Home > Th. List > pr2nelem | Structured version Visualization version GIF version |
Description: Lemma for pr2ne 8711. (Contributed by FL, 17-Aug-2008.) |
Ref | Expression |
---|---|
pr2nelem | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjsn2 4193 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
2 | ensn1g 7907 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1𝑜) | |
3 | ensn1g 7907 | . . . . 5 ⊢ (𝐵 ∈ 𝐷 → {𝐵} ≈ 1𝑜) | |
4 | pm54.43 8709 | . . . . . . 7 ⊢ (({𝐴} ≈ 1𝑜 ∧ {𝐵} ≈ 1𝑜) → (({𝐴} ∩ {𝐵}) = ∅ ↔ ({𝐴} ∪ {𝐵}) ≈ 2𝑜)) | |
5 | df-pr 4128 | . . . . . . . 8 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
6 | 5 | breq1i 4590 | . . . . . . 7 ⊢ ({𝐴, 𝐵} ≈ 2𝑜 ↔ ({𝐴} ∪ {𝐵}) ≈ 2𝑜) |
7 | 4, 6 | syl6bbr 277 | . . . . . 6 ⊢ (({𝐴} ≈ 1𝑜 ∧ {𝐵} ≈ 1𝑜) → (({𝐴} ∩ {𝐵}) = ∅ ↔ {𝐴, 𝐵} ≈ 2𝑜)) |
8 | 7 | biimpd 218 | . . . . 5 ⊢ (({𝐴} ≈ 1𝑜 ∧ {𝐵} ≈ 1𝑜) → (({𝐴} ∩ {𝐵}) = ∅ → {𝐴, 𝐵} ≈ 2𝑜)) |
9 | 2, 3, 8 | syl2an 493 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (({𝐴} ∩ {𝐵}) = ∅ → {𝐴, 𝐵} ≈ 2𝑜)) |
10 | 9 | ex 449 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (({𝐴} ∩ {𝐵}) = ∅ → {𝐴, 𝐵} ≈ 2𝑜))) |
11 | 1, 10 | syl7 72 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} ≈ 2𝑜))) |
12 | 11 | 3imp 1249 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 {cpr 4127 class class class wbr 4583 1𝑜c1o 7440 2𝑜c2o 7441 ≈ cen 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-2o 7448 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: pr2ne 8711 en2eqpr 8713 en2eleq 8714 pr2pwpr 13116 pmtrprfv 17696 pmtrprfv3 17697 symggen 17713 pmtr3ncomlem1 17716 pmtr3ncom 17718 mdetralt 20233 en2top 20600 hmphindis 21410 pmtrto1cl 29180 |
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