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Mirrors > Home > MPE Home > Th. List > pr2nelemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of enpr2 9938 as of 30-Dec-2024. (Contributed by FL, 17-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pr2nelemOLD | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjsn2 4673 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
2 | ensn1g 8963 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
3 | ensn1g 8963 | . . . . 5 ⊢ (𝐵 ∈ 𝐷 → {𝐵} ≈ 1o) | |
4 | pm54.43 9937 | . . . . . . 7 ⊢ (({𝐴} ≈ 1o ∧ {𝐵} ≈ 1o) → (({𝐴} ∩ {𝐵}) = ∅ ↔ ({𝐴} ∪ {𝐵}) ≈ 2o)) | |
5 | df-pr 4589 | . . . . . . . 8 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
6 | 5 | breq1i 5112 | . . . . . . 7 ⊢ ({𝐴, 𝐵} ≈ 2o ↔ ({𝐴} ∪ {𝐵}) ≈ 2o) |
7 | 4, 6 | bitr4di 288 | . . . . . 6 ⊢ (({𝐴} ≈ 1o ∧ {𝐵} ≈ 1o) → (({𝐴} ∩ {𝐵}) = ∅ ↔ {𝐴, 𝐵} ≈ 2o)) |
8 | 7 | biimpd 228 | . . . . 5 ⊢ (({𝐴} ≈ 1o ∧ {𝐵} ≈ 1o) → (({𝐴} ∩ {𝐵}) = ∅ → {𝐴, 𝐵} ≈ 2o)) |
9 | 2, 3, 8 | syl2an 596 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (({𝐴} ∩ {𝐵}) = ∅ → {𝐴, 𝐵} ≈ 2o)) |
10 | 9 | ex 413 | . . 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 1111 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∪ cun 3908 ∩ cin 3909 ∅c0 4282 {csn 4586 {cpr 4588 class class class wbr 5105 1oc1o 8405 2oc2o 8406 ≈ cen 8880 |
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-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-ord 6320 df-on 6321 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-1o 8412 df-2o 8413 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 |
This theorem is referenced by: (None) |
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