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Mirrors > Home > MPE Home > Th. List > ttukeyg | Structured version Visualization version GIF version |
Description: The Teichmüller-Tukey Lemma ttukey 9928 stated with the "choice" as an antecedent (the hypothesis ∪ 𝐴 ∈ dom card says that ∪ 𝐴 is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
ttukeyg | ⊢ ((∪ 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4307 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴) | |
2 | ttukey2g 9926 | . . . . . 6 ⊢ ((∪ 𝐴 ∈ dom card ∧ 𝑧 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝑧 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)) | |
3 | simpr 485 | . . . . . . 7 ⊢ ((𝑧 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | |
4 | 3 | reximi 3240 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ ((∪ 𝐴 ∈ dom card ∧ 𝑧 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) |
6 | 5 | 3exp 1111 | . . . 4 ⊢ (∪ 𝐴 ∈ dom card → (𝑧 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))) |
7 | 6 | exlimdv 1925 | . . 3 ⊢ (∪ 𝐴 ∈ dom card → (∃𝑧 𝑧 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))) |
8 | 1, 7 | syl5bi 243 | . 2 ⊢ (∪ 𝐴 ∈ dom card → (𝐴 ≠ ∅ → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))) |
9 | 8 | 3imp 1103 | 1 ⊢ ((∪ 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 ∀wal 1526 ∃wex 1771 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 ∩ cin 3932 ⊆ wss 3933 ⊊ wpss 3934 ∅c0 4288 𝒫 cpw 4535 ∪ cuni 4830 dom cdm 5548 Fincfn 8497 cardccrd 9352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-om 7570 df-wrecs 7936 df-recs 7997 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-fin 8501 df-card 9356 |
This theorem is referenced by: ttukey 9928 |
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