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Mirrors > Home > MPE Home > Th. List > pr2nelem | Structured version Visualization version GIF version |
Description: Lemma for pr2ne 8828. (Contributed by FL, 17-Aug-2008.) |
Ref | Expression |
---|---|
pr2nelem | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjsn2 4247 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
2 | ensn1g 8021 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1𝑜) | |
3 | ensn1g 8021 | . . . . 5 ⊢ (𝐵 ∈ 𝐷 → {𝐵} ≈ 1𝑜) | |
4 | pm54.43 8826 | . . . . . . 7 ⊢ (({𝐴} ≈ 1𝑜 ∧ {𝐵} ≈ 1𝑜) → (({𝐴} ∩ {𝐵}) = ∅ ↔ ({𝐴} ∪ {𝐵}) ≈ 2𝑜)) | |
5 | df-pr 4180 | . . . . . . . 8 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
6 | 5 | breq1i 4660 | . . . . . . 7 ⊢ ({𝐴, 𝐵} ≈ 2𝑜 ↔ ({𝐴} ∪ {𝐵}) ≈ 2𝑜) |
7 | 4, 6 | syl6bbr 278 | . . . . . 6 ⊢ (({𝐴} ≈ 1𝑜 ∧ {𝐵} ≈ 1𝑜) → (({𝐴} ∩ {𝐵}) = ∅ ↔ {𝐴, 𝐵} ≈ 2𝑜)) |
8 | 7 | biimpd 219 | . . . . 5 ⊢ (({𝐴} ≈ 1𝑜 ∧ {𝐵} ≈ 1𝑜) → (({𝐴} ∩ {𝐵}) = ∅ → {𝐴, 𝐵} ≈ 2𝑜)) |
9 | 2, 3, 8 | syl2an 494 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (({𝐴} ∩ {𝐵}) = ∅ → {𝐴, 𝐵} ≈ 2𝑜)) |
10 | 9 | ex 450 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (({𝐴} ∩ {𝐵}) = ∅ → {𝐴, 𝐵} ≈ 2𝑜))) |
11 | 1, 10 | syl7 74 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} ≈ 2𝑜))) |
12 | 11 | 3imp 1256 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∪ cun 3572 ∩ cin 3573 ∅c0 3915 {csn 4177 {cpr 4179 class class class wbr 4653 1𝑜c1o 7553 2𝑜c2o 7554 ≈ cen 7952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 |
This theorem is referenced by: pr2ne 8828 en2eqpr 8830 en2eleq 8831 pr2pwpr 13261 pmtrprfv 17873 pmtrprfv3 17874 symggen 17890 pmtr3ncomlem1 17893 pmtr3ncom 17895 mdetralt 20414 en2top 20789 hmphindis 21600 pmtrto1cl 29849 pmtridf1o 29856 |
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