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Mirrors > Home > MPE Home > Th. List > tskpr | Structured version Visualization version GIF version |
Description: If 𝐴 and 𝐵 are members of a Tarski class, their unordered pair is also an element of the class. JFM CLASSES2 th. 3 (partly). (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Jun-2013.) |
Ref | Expression |
---|---|
tskpr | ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1138 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → 𝑇 ∈ Tarski) | |
2 | prssi 4720 | . . 3 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ⊆ 𝑇) | |
3 | 2 | 3adant1 1132 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ⊆ 𝑇) |
4 | prfi 8924 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ Fin | |
5 | isfinite 9245 | . . . . 5 ⊢ ({𝐴, 𝐵} ∈ Fin ↔ {𝐴, 𝐵} ≺ ω) | |
6 | 4, 5 | mpbi 233 | . . . 4 ⊢ {𝐴, 𝐵} ≺ ω |
7 | ne0i 4235 | . . . . 5 ⊢ (𝐴 ∈ 𝑇 → 𝑇 ≠ ∅) | |
8 | tskinf 10348 | . . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇) | |
9 | 7, 8 | sylan2 596 | . . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → ω ≼ 𝑇) |
10 | sdomdomtr 8757 | . . . 4 ⊢ (({𝐴, 𝐵} ≺ ω ∧ ω ≼ 𝑇) → {𝐴, 𝐵} ≺ 𝑇) | |
11 | 6, 9, 10 | sylancr 590 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝐴, 𝐵} ≺ 𝑇) |
12 | 11 | 3adant3 1134 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ≺ 𝑇) |
13 | tskssel 10336 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ {𝐴, 𝐵} ⊆ 𝑇 ∧ {𝐴, 𝐵} ≺ 𝑇) → {𝐴, 𝐵} ∈ 𝑇) | |
14 | 1, 3, 12, 13 | syl3anc 1373 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2112 ≠ wne 2932 ⊆ wss 3853 ∅c0 4223 {cpr 4529 class class class wbr 5039 ωcom 7622 ≼ cdom 8602 ≺ csdm 8603 Fincfn 8604 Tarskictsk 10327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-r1 9345 df-tsk 10328 |
This theorem is referenced by: tskop 10350 tskwun 10363 tskun 10365 grutsk1 10400 |
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