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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 1128 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → 𝑇 ∈ Tarski) | |
2 | prssi 4746 | . . 3 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ⊆ 𝑇) | |
3 | 2 | 3adant1 1122 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ⊆ 𝑇) |
4 | prfi 8781 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ Fin | |
5 | isfinite 9103 | . . . . 5 ⊢ ({𝐴, 𝐵} ∈ Fin ↔ {𝐴, 𝐵} ≺ ω) | |
6 | 4, 5 | mpbi 231 | . . . 4 ⊢ {𝐴, 𝐵} ≺ ω |
7 | ne0i 4297 | . . . . 5 ⊢ (𝐴 ∈ 𝑇 → 𝑇 ≠ ∅) | |
8 | tskinf 10179 | . . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇) | |
9 | 7, 8 | sylan2 592 | . . . 4 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → ω ≼ 𝑇) |
10 | sdomdomtr 8638 | . . . 4 ⊢ (({𝐴, 𝐵} ≺ ω ∧ ω ≼ 𝑇) → {𝐴, 𝐵} ≺ 𝑇) | |
11 | 6, 9, 10 | sylancr 587 | . . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝐴, 𝐵} ≺ 𝑇) |
12 | 11 | 3adant3 1124 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ≺ 𝑇) |
13 | tskssel 10167 | . 2 ⊢ ((𝑇 ∈ Tarski ∧ {𝐴, 𝐵} ⊆ 𝑇 ∧ {𝐴, 𝐵} ≺ 𝑇) → {𝐴, 𝐵} ∈ 𝑇) | |
14 | 1, 3, 12, 13 | syl3anc 1363 | 1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 ≠ wne 3013 ⊆ wss 3933 ∅c0 4288 {cpr 4559 class class class wbr 5057 ωcom 7569 ≼ cdom 8495 ≺ csdm 8496 Fincfn 8497 Tarskictsk 10158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-r1 9181 df-tsk 10159 |
This theorem is referenced by: tskop 10181 tskwun 10194 tskun 10196 grutsk1 10231 |
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