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Mirrors > Home > MPE Home > Th. List > tskr1om | Structured version Visualization version GIF version |
Description: A nonempty Tarski class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 9087.) (Contributed by Mario Carneiro, 24-Jun-2013.) |
Ref | Expression |
---|---|
tskr1om | ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6656 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅)) | |
2 | 1 | eleq1d 2897 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘∅) ∈ 𝑇)) |
3 | fveq2 6656 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑅1‘𝑥) = (𝑅1‘𝑦)) | |
4 | 3 | eleq1d 2897 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘𝑦) ∈ 𝑇)) |
5 | fveq2 6656 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑅1‘𝑥) = (𝑅1‘suc 𝑦)) | |
6 | 5 | eleq1d 2897 | . . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ (𝑅1‘suc 𝑦) ∈ 𝑇)) |
7 | r10 9183 | . . . . . . 7 ⊢ (𝑅1‘∅) = ∅ | |
8 | tsk0 10171 | . . . . . . 7 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇) | |
9 | 7, 8 | eqeltrid 2917 | . . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘∅) ∈ 𝑇) |
10 | tskpw 10161 | . . . . . . . . 9 ⊢ ((𝑇 ∈ Tarski ∧ (𝑅1‘𝑦) ∈ 𝑇) → 𝒫 (𝑅1‘𝑦) ∈ 𝑇) | |
11 | nnon 7572 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On) | |
12 | r1suc 9185 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦)) | |
13 | 11, 12 | syl 17 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1‘𝑦)) |
14 | 13 | eleq1d 2897 | . . . . . . . . 9 ⊢ (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑇 ↔ 𝒫 (𝑅1‘𝑦) ∈ 𝑇)) |
15 | 10, 14 | syl5ibr 248 | . . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ (𝑅1‘𝑦) ∈ 𝑇) → (𝑅1‘suc 𝑦) ∈ 𝑇)) |
16 | 15 | expd 418 | . . . . . . 7 ⊢ (𝑦 ∈ ω → (𝑇 ∈ Tarski → ((𝑅1‘𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇))) |
17 | 16 | adantrd 494 | . . . . . 6 ⊢ (𝑦 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ((𝑅1‘𝑦) ∈ 𝑇 → (𝑅1‘suc 𝑦) ∈ 𝑇))) |
18 | 2, 4, 6, 9, 17 | finds2 7596 | . . . . 5 ⊢ (𝑥 ∈ ω → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘𝑥) ∈ 𝑇)) |
19 | eleq1 2900 | . . . . . 6 ⊢ ((𝑅1‘𝑥) = 𝑦 → ((𝑅1‘𝑥) ∈ 𝑇 ↔ 𝑦 ∈ 𝑇)) | |
20 | 19 | imbi2d 343 | . . . . 5 ⊢ ((𝑅1‘𝑥) = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1‘𝑥) ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇))) |
21 | 18, 20 | syl5ibcom 247 | . . . 4 ⊢ (𝑥 ∈ ω → ((𝑅1‘𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇))) |
22 | 21 | rexlimiv 3280 | . . 3 ⊢ (∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑦 ∈ 𝑇)) |
23 | r1fnon 9182 | . . . . 5 ⊢ 𝑅1 Fn On | |
24 | fnfun 6439 | . . . . 5 ⊢ (𝑅1 Fn On → Fun 𝑅1) | |
25 | 23, 24 | ax-mp 5 | . . . 4 ⊢ Fun 𝑅1 |
26 | fvelima 6717 | . . . 4 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦) | |
27 | 25, 26 | mpan 688 | . . 3 ⊢ (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1‘𝑥) = 𝑦) |
28 | 22, 27 | syl11 33 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑦 ∈ (𝑅1 “ ω) → 𝑦 ∈ 𝑇)) |
29 | 28 | ssrdv 3961 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ⊆ wss 3924 ∅c0 4279 𝒫 cpw 4525 “ cima 5544 Oncon0 6177 suc csuc 6179 Fun wfun 6335 Fn wfn 6336 ‘cfv 6341 ωcom 7566 𝑅1cr1 9177 Tarskictsk 10156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-om 7567 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-r1 9179 df-tsk 10157 |
This theorem is referenced by: tskr1om2 10176 tskinf 10177 |
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