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Mirrors > Home > MPE Home > Th. List > tskr1om2 | Structured version Visualization version GIF version |
Description: A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 9575.) (Contributed by NM, 22-Feb-2011.) |
Ref | Expression |
---|---|
tskr1om2 | ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∪ (𝑅1 “ ω) ⊆ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni2 4870 | . . 3 ⊢ (𝑦 ∈ ∪ (𝑅1 “ ω) ↔ ∃𝑥 ∈ (𝑅1 “ ω)𝑦 ∈ 𝑥) | |
2 | r1fnon 9704 | . . . . . . . . 9 ⊢ 𝑅1 Fn On | |
3 | fnfun 6603 | . . . . . . . . 9 ⊢ (𝑅1 Fn On → Fun 𝑅1) | |
4 | 2, 3 | ax-mp 5 | . . . . . . . 8 ⊢ Fun 𝑅1 |
5 | fvelima 6909 | . . . . . . . 8 ⊢ ((Fun 𝑅1 ∧ 𝑥 ∈ (𝑅1 “ ω)) → ∃𝑦 ∈ ω (𝑅1‘𝑦) = 𝑥) | |
6 | 4, 5 | mpan 689 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑅1 “ ω) → ∃𝑦 ∈ ω (𝑅1‘𝑦) = 𝑥) |
7 | r1tr 9713 | . . . . . . . . 9 ⊢ Tr (𝑅1‘𝑦) | |
8 | treq 5231 | . . . . . . . . 9 ⊢ ((𝑅1‘𝑦) = 𝑥 → (Tr (𝑅1‘𝑦) ↔ Tr 𝑥)) | |
9 | 7, 8 | mpbii 232 | . . . . . . . 8 ⊢ ((𝑅1‘𝑦) = 𝑥 → Tr 𝑥) |
10 | 9 | rexlimivw 3149 | . . . . . . 7 ⊢ (∃𝑦 ∈ ω (𝑅1‘𝑦) = 𝑥 → Tr 𝑥) |
11 | trss 5234 | . . . . . . 7 ⊢ (Tr 𝑥 → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥)) | |
12 | 6, 10, 11 | 3syl 18 | . . . . . 6 ⊢ (𝑥 ∈ (𝑅1 “ ω) → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥)) |
13 | 12 | adantl 483 | . . . . 5 ⊢ (((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ (𝑅1 “ ω)) → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥)) |
14 | tskr1om 10704 | . . . . . . . 8 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇) | |
15 | 14 | sseld 3944 | . . . . . . 7 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑥 ∈ (𝑅1 “ ω) → 𝑥 ∈ 𝑇)) |
16 | tskss 10695 | . . . . . . . . 9 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇 ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ 𝑇) | |
17 | 16 | 3exp 1120 | . . . . . . . 8 ⊢ (𝑇 ∈ Tarski → (𝑥 ∈ 𝑇 → (𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝑇))) |
18 | 17 | adantr 482 | . . . . . . 7 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑥 ∈ 𝑇 → (𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝑇))) |
19 | 15, 18 | syld 47 | . . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑥 ∈ (𝑅1 “ ω) → (𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝑇))) |
20 | 19 | imp 408 | . . . . 5 ⊢ (((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ (𝑅1 “ ω)) → (𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝑇)) |
21 | 13, 20 | syld 47 | . . . 4 ⊢ (((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ (𝑅1 “ ω)) → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑇)) |
22 | 21 | rexlimdva 3153 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (∃𝑥 ∈ (𝑅1 “ ω)𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑇)) |
23 | 1, 22 | biimtrid 241 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑦 ∈ ∪ (𝑅1 “ ω) → 𝑦 ∈ 𝑇)) |
24 | 23 | ssrdv 3951 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∪ (𝑅1 “ ω) ⊆ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∃wrex 3074 ⊆ wss 3911 ∅c0 4283 ∪ cuni 4866 Tr wtr 5223 “ cima 5637 Oncon0 6318 Fun wfun 6491 Fn wfn 6492 ‘cfv 6497 ωcom 7803 𝑅1cr1 9699 Tarskictsk 10685 |
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 |
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-r1 9701 df-tsk 10686 |
This theorem is referenced by: (None) |
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