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Mirrors > Home > MPE Home > Th. List > Mathboxes > prct | Structured version Visualization version GIF version |
Description: An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.) |
Ref | Expression |
---|---|
prct | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4563 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | snct 30443 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≼ ω) | |
3 | snct 30443 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ≼ ω) | |
4 | unctb 9621 | . . 3 ⊢ (({𝐴} ≼ ω ∧ {𝐵} ≼ ω) → ({𝐴} ∪ {𝐵}) ≼ ω) | |
5 | 2, 3, 4 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ∪ {𝐵}) ≼ ω) |
6 | 1, 5 | eqbrtrid 5093 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ∪ cun 3933 {csn 4560 {cpr 4562 class class class wbr 5058 ωcom 7574 ≼ cdom 8501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-oi 8968 df-dju 9324 df-card 9362 |
This theorem is referenced by: difelsiga 31387 unelsiga 31388 unelldsys 31412 measxun2 31464 measssd 31469 carsgsigalem 31568 carsgclctun 31574 pmeasmono 31577 probun 31672 |
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