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Mirrors > Home > MPE Home > Th. List > tskwun | Structured version Visualization version GIF version |
Description: A nonempty transitive Tarski class is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
tskwun | ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → 𝑇 ∈ WUni) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1062 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → Tr 𝑇) | |
2 | simp3 1063 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → 𝑇 ≠ ∅) | |
3 | tskuni 9605 | . . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑥 ∈ 𝑇) → ∪ 𝑥 ∈ 𝑇) | |
4 | 3 | 3expa 1265 | . . . . 5 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥 ∈ 𝑇) → ∪ 𝑥 ∈ 𝑇) |
5 | 4 | 3adantl3 1219 | . . . 4 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ 𝑇) → ∪ 𝑥 ∈ 𝑇) |
6 | tskpw 9575 | . . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇) → 𝒫 𝑥 ∈ 𝑇) | |
7 | 6 | 3ad2antl1 1223 | . . . 4 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ 𝑇) → 𝒫 𝑥 ∈ 𝑇) |
8 | tskpr 9592 | . . . . . . . 8 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇) → {𝑥, 𝑦} ∈ 𝑇) | |
9 | 8 | 3exp 1264 | . . . . . . 7 ⊢ (𝑇 ∈ Tarski → (𝑥 ∈ 𝑇 → (𝑦 ∈ 𝑇 → {𝑥, 𝑦} ∈ 𝑇))) |
10 | 9 | 3ad2ant1 1082 | . . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → (𝑥 ∈ 𝑇 → (𝑦 ∈ 𝑇 → {𝑥, 𝑦} ∈ 𝑇))) |
11 | 10 | imp31 448 | . . . . 5 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → {𝑥, 𝑦} ∈ 𝑇) |
12 | 11 | ralrimiva 2966 | . . . 4 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ 𝑇) → ∀𝑦 ∈ 𝑇 {𝑥, 𝑦} ∈ 𝑇) |
13 | 5, 7, 12 | 3jca 1242 | . . 3 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ 𝑇) → (∪ 𝑥 ∈ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {𝑥, 𝑦} ∈ 𝑇)) |
14 | 13 | ralrimiva 2966 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → ∀𝑥 ∈ 𝑇 (∪ 𝑥 ∈ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {𝑥, 𝑦} ∈ 𝑇)) |
15 | iswun 9526 | . . 3 ⊢ (𝑇 ∈ Tarski → (𝑇 ∈ WUni ↔ (Tr 𝑇 ∧ 𝑇 ≠ ∅ ∧ ∀𝑥 ∈ 𝑇 (∪ 𝑥 ∈ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {𝑥, 𝑦} ∈ 𝑇)))) | |
16 | 15 | 3ad2ant1 1082 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → (𝑇 ∈ WUni ↔ (Tr 𝑇 ∧ 𝑇 ≠ ∅ ∧ ∀𝑥 ∈ 𝑇 (∪ 𝑥 ∈ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {𝑥, 𝑦} ∈ 𝑇)))) |
17 | 1, 2, 14, 16 | mpbir3and 1245 | 1 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → 𝑇 ∈ WUni) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∅c0 3915 𝒫 cpw 4158 {cpr 4179 ∪ cuni 4436 Tr wtr 4752 WUnicwun 9522 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 ax-ac2 9285 |
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-rmo 2920 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-int 4476 df-iun 4522 df-iin 4523 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-smo 7443 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-oi 8415 df-har 8463 df-r1 8627 df-card 8765 df-aleph 8766 df-cf 8767 df-acn 8768 df-ac 8939 df-wina 9506 df-ina 9507 df-wun 9524 df-tsk 9571 |
This theorem is referenced by: tskxp 9609 tskmap 9610 |
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