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Mirrors > Home > MPE Home > Th. List > tskinf | Structured version Visualization version GIF version |
Description: A nonempty Tarski class is infinite. (Contributed by FL, 22-Feb-2011.) |
Ref | Expression |
---|---|
tskinf | ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r111 8997 | . . . 4 ⊢ 𝑅1:On–1-1→V | |
2 | omsson 7399 | . . . 4 ⊢ ω ⊆ On | |
3 | omex 8899 | . . . . 5 ⊢ ω ∈ V | |
4 | 3 | f1imaen 8367 | . . . 4 ⊢ ((𝑅1:On–1-1→V ∧ ω ⊆ On) → (𝑅1 “ ω) ≈ ω) |
5 | 1, 2, 4 | mp2an 680 | . . 3 ⊢ (𝑅1 “ ω) ≈ ω |
6 | 5 | ensymi 8355 | . 2 ⊢ ω ≈ (𝑅1 “ ω) |
7 | simpl 475 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑇 ∈ Tarski) | |
8 | tskr1om 9986 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇) | |
9 | ssdomg 8351 | . . 3 ⊢ (𝑇 ∈ Tarski → ((𝑅1 “ ω) ⊆ 𝑇 → (𝑅1 “ ω) ≼ 𝑇)) | |
10 | 7, 8, 9 | sylc 65 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ≼ 𝑇) |
11 | endomtr 8363 | . 2 ⊢ ((ω ≈ (𝑅1 “ ω) ∧ (𝑅1 “ ω) ≼ 𝑇) → ω ≼ 𝑇) | |
12 | 6, 10, 11 | sylancr 579 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2051 ≠ wne 2962 Vcvv 3410 ⊆ wss 3824 ∅c0 4173 class class class wbr 4926 “ cima 5407 Oncon0 6027 –1-1→wf1 6183 ωcom 7395 ≈ cen 8302 ≼ cdom 8303 𝑅1cr1 8984 Tarskictsk 9967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-inf2 8897 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-om 7396 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-r1 8986 df-tsk 9968 |
This theorem is referenced by: tskpr 9989 |
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