Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 9198 | . . . 4 ⊢ 𝑅1:On–1-1→V | |
2 | omsson 7578 | . . . 4 ⊢ ω ⊆ On | |
3 | omex 9100 | . . . . 5 ⊢ ω ∈ V | |
4 | 3 | f1imaen 8565 | . . . 4 ⊢ ((𝑅1:On–1-1→V ∧ ω ⊆ On) → (𝑅1 “ ω) ≈ ω) |
5 | 1, 2, 4 | mp2an 690 | . . 3 ⊢ (𝑅1 “ ω) ≈ ω |
6 | 5 | ensymi 8553 | . 2 ⊢ ω ≈ (𝑅1 “ ω) |
7 | simpl 485 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑇 ∈ Tarski) | |
8 | tskr1om 10183 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇) | |
9 | ssdomg 8549 | . . 3 ⊢ (𝑇 ∈ Tarski → ((𝑅1 “ ω) ⊆ 𝑇 → (𝑅1 “ ω) ≼ 𝑇)) | |
10 | 7, 8, 9 | sylc 65 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ≼ 𝑇) |
11 | endomtr 8561 | . 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 2110 ≠ wne 3016 Vcvv 3495 ⊆ wss 3936 ∅c0 4291 class class class wbr 5059 “ cima 5553 Oncon0 6186 –1-1→wf1 6347 ωcom 7574 ≈ cen 8500 ≼ cdom 8501 𝑅1cr1 9185 Tarskictsk 10164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-r1 9187 df-tsk 10165 |
This theorem is referenced by: tskpr 10186 |
Copyright terms: Public domain | W3C validator |