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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 9204 | . . . 4 ⊢ 𝑅1:On–1-1→V | |
2 | omsson 7584 | . . . 4 ⊢ ω ⊆ On | |
3 | omex 9106 | . . . . 5 ⊢ ω ∈ V | |
4 | 3 | f1imaen 8571 | . . . 4 ⊢ ((𝑅1:On–1-1→V ∧ ω ⊆ On) → (𝑅1 “ ω) ≈ ω) |
5 | 1, 2, 4 | mp2an 690 | . . 3 ⊢ (𝑅1 “ ω) ≈ ω |
6 | 5 | ensymi 8559 | . 2 ⊢ ω ≈ (𝑅1 “ ω) |
7 | simpl 485 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑇 ∈ Tarski) | |
8 | tskr1om 10189 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇) | |
9 | ssdomg 8555 | . . 3 ⊢ (𝑇 ∈ Tarski → ((𝑅1 “ ω) ⊆ 𝑇 → (𝑅1 “ ω) ≼ 𝑇)) | |
10 | 7, 8, 9 | sylc 65 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ≼ 𝑇) |
11 | endomtr 8567 | . 2 ⊢ ((ω ≈ (𝑅1 “ ω) ∧ (𝑅1 “ ω) ≼ 𝑇) → ω ≼ 𝑇) | |
12 | 6, 10, 11 | sylancr 589 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 “ cima 5558 Oncon0 6191 –1-1→wf1 6352 ωcom 7580 ≈ cen 8506 ≼ cdom 8507 𝑅1cr1 9191 Tarskictsk 10170 |
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 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-r1 9193 df-tsk 10171 |
This theorem is referenced by: tskpr 10192 |
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