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 8638 | . . . 4 ⊢ 𝑅1:On–1-1→V | |
2 | omsson 7069 | . . . 4 ⊢ ω ⊆ On | |
3 | omex 8540 | . . . . 5 ⊢ ω ∈ V | |
4 | 3 | f1imaen 8018 | . . . 4 ⊢ ((𝑅1:On–1-1→V ∧ ω ⊆ On) → (𝑅1 “ ω) ≈ ω) |
5 | 1, 2, 4 | mp2an 708 | . . 3 ⊢ (𝑅1 “ ω) ≈ ω |
6 | 5 | ensymi 8006 | . 2 ⊢ ω ≈ (𝑅1 “ ω) |
7 | simpl 473 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 𝑇 ∈ Tarski) | |
8 | tskr1om 9589 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇) | |
9 | ssdomg 8001 | . . 3 ⊢ (𝑇 ∈ Tarski → ((𝑅1 “ ω) ⊆ 𝑇 → (𝑅1 “ ω) ≼ 𝑇)) | |
10 | 7, 8, 9 | sylc 65 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ≼ 𝑇) |
11 | endomtr 8014 | . 2 ⊢ ((ω ≈ (𝑅1 “ ω) ∧ (𝑅1 “ ω) ≼ 𝑇) → ω ≼ 𝑇) | |
12 | 6, 10, 11 | sylancr 695 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 “ cima 5117 Oncon0 5723 –1-1→wf1 5885 ωcom 7065 ≈ cen 7952 ≼ cdom 7953 𝑅1cr1 8625 Tarskictsk 9570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-r1 8627 df-tsk 9571 |
This theorem is referenced by: tskpr 9592 |
Copyright terms: Public domain | W3C validator |