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Mirrors > Home > MPE Home > Th. List > zorn2lem3 | Structured version Visualization version GIF version |
Description: Lemma for zorn2 9328. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
Ref | Expression |
---|---|
zorn2lem.3 | ⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))) |
zorn2lem.4 | ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧} |
zorn2lem.5 | ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧} |
Ref | Expression |
---|---|
zorn2lem3 | ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅))) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zorn2lem.3 | . . . 4 ⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))) | |
2 | zorn2lem.4 | . . . 4 ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧} | |
3 | zorn2lem.5 | . . . 4 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧} | |
4 | 1, 2, 3 | zorn2lem2 9319 | . . 3 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝑦 ∈ 𝑥 → (𝐹‘𝑦)𝑅(𝐹‘𝑥))) |
5 | 4 | adantl 482 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅))) → (𝑦 ∈ 𝑥 → (𝐹‘𝑦)𝑅(𝐹‘𝑥))) |
6 | ssrab2 3687 | . . . . 5 ⊢ {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧} ⊆ 𝐴 | |
7 | 3, 6 | eqsstri 3635 | . . . 4 ⊢ 𝐷 ⊆ 𝐴 |
8 | 1, 2, 3 | zorn2lem1 9318 | . . . 4 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷) |
9 | 7, 8 | sseldi 3601 | . . 3 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐴) |
10 | breq1 4656 | . . . . . 6 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → ((𝐹‘𝑥)𝑅(𝐹‘𝑥) ↔ (𝐹‘𝑦)𝑅(𝐹‘𝑥))) | |
11 | 10 | biimprcd 240 | . . . . 5 ⊢ ((𝐹‘𝑦)𝑅(𝐹‘𝑥) → ((𝐹‘𝑥) = (𝐹‘𝑦) → (𝐹‘𝑥)𝑅(𝐹‘𝑥))) |
12 | poirr 5046 | . . . . 5 ⊢ ((𝑅 Po 𝐴 ∧ (𝐹‘𝑥) ∈ 𝐴) → ¬ (𝐹‘𝑥)𝑅(𝐹‘𝑥)) | |
13 | 11, 12 | nsyli 155 | . . . 4 ⊢ ((𝐹‘𝑦)𝑅(𝐹‘𝑥) → ((𝑅 Po 𝐴 ∧ (𝐹‘𝑥) ∈ 𝐴) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) |
14 | 13 | com12 32 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ (𝐹‘𝑥) ∈ 𝐴) → ((𝐹‘𝑦)𝑅(𝐹‘𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) |
15 | 9, 14 | sylan2 491 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅))) → ((𝐹‘𝑦)𝑅(𝐹‘𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) |
16 | 5, 15 | syld 47 | 1 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅))) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 Vcvv 3200 ∅c0 3915 class class class wbr 4653 ↦ cmpt 4729 Po wpo 5033 We wwe 5072 ran crn 5115 “ cima 5117 Oncon0 5723 ‘cfv 5888 ℩crio 6610 recscrecs 7467 |
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-rep 4771 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-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 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-pred 5680 df-ord 5726 df-on 5727 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-riota 6611 df-wrecs 7407 df-recs 7468 |
This theorem is referenced by: zorn2lem4 9321 |
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