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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 4628 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | snct 32490 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≼ ω) | |
3 | snct 32490 | . . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ≼ ω) | |
4 | unctb 10223 | . . 3 ⊢ (({𝐴} ≼ ω ∧ {𝐵} ≼ ω) → ({𝐴} ∪ {𝐵}) ≼ ω) | |
5 | 2, 3, 4 | syl2an 595 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ∪ {𝐵}) ≼ ω) |
6 | 1, 5 | eqbrtrid 5178 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ∪ cun 3943 {csn 4625 {cpr 4627 class class class wbr 5143 ωcom 7865 ≼ cdom 8956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7371 df-ov 7418 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-2o 8482 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-oi 9528 df-dju 9919 df-card 9957 |
This theorem is referenced by: difelsiga 33747 unelsiga 33748 unelldsys 33772 measxun2 33824 measssd 33829 carsgsigalem 33930 carsgclctun 33936 pmeasmono 33939 probun 34034 |
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