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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 1130 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → Tr 𝑇) | |
2 | simp3 1131 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → 𝑇 ≠ ∅) | |
3 | tskuni 10051 | . . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑥 ∈ 𝑇) → ∪ 𝑥 ∈ 𝑇) | |
4 | 3 | 3expa 1111 | . . . . 5 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝑥 ∈ 𝑇) → ∪ 𝑥 ∈ 𝑇) |
5 | 4 | 3adantl3 1161 | . . . 4 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ 𝑇) → ∪ 𝑥 ∈ 𝑇) |
6 | tskpw 10021 | . . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇) → 𝒫 𝑥 ∈ 𝑇) | |
7 | 6 | 3ad2antl1 1178 | . . . 4 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ 𝑇) → 𝒫 𝑥 ∈ 𝑇) |
8 | tskpr 10038 | . . . . . . . 8 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇) → {𝑥, 𝑦} ∈ 𝑇) | |
9 | 8 | 3exp 1112 | . . . . . . 7 ⊢ (𝑇 ∈ Tarski → (𝑥 ∈ 𝑇 → (𝑦 ∈ 𝑇 → {𝑥, 𝑦} ∈ 𝑇))) |
10 | 9 | 3ad2ant1 1126 | . . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → (𝑥 ∈ 𝑇 → (𝑦 ∈ 𝑇 → {𝑥, 𝑦} ∈ 𝑇))) |
11 | 10 | imp31 418 | . . . . 5 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → {𝑥, 𝑦} ∈ 𝑇) |
12 | 11 | ralrimiva 3149 | . . . 4 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ 𝑇) → ∀𝑦 ∈ 𝑇 {𝑥, 𝑦} ∈ 𝑇) |
13 | 5, 7, 12 | 3jca 1121 | . . 3 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) ∧ 𝑥 ∈ 𝑇) → (∪ 𝑥 ∈ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {𝑥, 𝑦} ∈ 𝑇)) |
14 | 13 | ralrimiva 3149 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → ∀𝑥 ∈ 𝑇 (∪ 𝑥 ∈ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {𝑥, 𝑦} ∈ 𝑇)) |
15 | iswun 9972 | . . 3 ⊢ (𝑇 ∈ Tarski → (𝑇 ∈ WUni ↔ (Tr 𝑇 ∧ 𝑇 ≠ ∅ ∧ ∀𝑥 ∈ 𝑇 (∪ 𝑥 ∈ 𝑇 ∧ 𝒫 𝑥 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {𝑥, 𝑦} ∈ 𝑇)))) | |
16 | 15 | 3ad2ant1 1126 | . 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 1335 | 1 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → 𝑇 ∈ WUni) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 ∅c0 4211 𝒫 cpw 4453 {cpr 4474 ∪ cuni 4745 Tr wtr 5063 WUnicwun 9968 Tarskictsk 10016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-ac2 9731 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-smo 7835 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-oi 8820 df-har 8868 df-r1 9039 df-card 9214 df-aleph 9215 df-cf 9216 df-acn 9217 df-ac 9388 df-wina 9952 df-ina 9953 df-wun 9970 df-tsk 10017 |
This theorem is referenced by: tskxp 10055 tskmap 10056 |
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