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Mirrors > Home > MPE Home > Th. List > axdc | Structured version Visualization version GIF version |
Description: This theorem derives ax-dc 10383 using ax-ac 10396 and ax-inf 9575. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (New usage is discouraged.) (Contributed by Mario Carneiro, 25-Jan-2013.) |
Ref | Expression |
---|---|
axdc | ⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5110 | . . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑢𝑥𝑤 ↔ 𝑢𝑥𝑧)) | |
2 | 1 | cbvabv 2810 | . . . . . . . 8 ⊢ {𝑤 ∣ 𝑢𝑥𝑤} = {𝑧 ∣ 𝑢𝑥𝑧} |
3 | breq1 5109 | . . . . . . . . 9 ⊢ (𝑢 = 𝑣 → (𝑢𝑥𝑧 ↔ 𝑣𝑥𝑧)) | |
4 | 3 | abbidv 2806 | . . . . . . . 8 ⊢ (𝑢 = 𝑣 → {𝑧 ∣ 𝑢𝑥𝑧} = {𝑧 ∣ 𝑣𝑥𝑧}) |
5 | 2, 4 | eqtrid 2789 | . . . . . . 7 ⊢ (𝑢 = 𝑣 → {𝑤 ∣ 𝑢𝑥𝑤} = {𝑧 ∣ 𝑣𝑥𝑧}) |
6 | 5 | fveq2d 6847 | . . . . . 6 ⊢ (𝑢 = 𝑣 → (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤}) = (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})) |
7 | 6 | cbvmptv 5219 | . . . . 5 ⊢ (𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})) |
8 | rdgeq1 8358 | . . . . 5 ⊢ ((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})) = (𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})) → rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) = rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦)) | |
9 | reseq1 5932 | . . . . 5 ⊢ (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) = rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦) → (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) ↾ ω) = (rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦) ↾ ω)) | |
10 | 7, 8, 9 | mp2b 10 | . . . 4 ⊢ (rec((𝑢 ∈ V ↦ (𝑔‘{𝑤 ∣ 𝑢𝑥𝑤})), 𝑦) ↾ ω) = (rec((𝑣 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑣𝑥𝑧})), 𝑦) ↾ ω) |
11 | 10 | axdclem2 10457 | . . 3 ⊢ (∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))) |
12 | 11 | exlimiv 1934 | . 2 ⊢ (∃𝑦∃𝑧 𝑦𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))) |
13 | 12 | imp 408 | 1 ⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 {cab 2714 ∀wral 3065 Vcvv 3446 ⊆ wss 3911 class class class wbr 5106 ↦ cmpt 5189 dom cdm 5634 ran crn 5635 ↾ cres 5636 suc csuc 6320 ‘cfv 6497 ωcom 7803 reccrdg 8356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 ax-ac2 10400 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-ac 10053 |
This theorem is referenced by: (None) |
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