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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunctb2 | Structured version Visualization version GIF version |
Description: Using the axiom of countable choice ax-cc 10500, the countable union of countable sets is countable. See iunctb 10639 for a somewhat more general theorem. (Contributed by ML, 10-Dec-2020.) |
Ref | Expression |
---|---|
iunctb2 | ⊢ (∀𝑥 ∈ ω 𝐵 ≼ ω → ∪ 𝑥 ∈ ω 𝐵 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 9708 | . . 3 ⊢ ω ∈ V | |
2 | domrefg 9043 | . . 3 ⊢ (ω ∈ V → ω ≼ ω) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ω ≼ ω |
4 | iunctb 10639 | . 2 ⊢ ((ω ≼ ω ∧ ∀𝑥 ∈ ω 𝐵 ≼ ω) → ∪ 𝑥 ∈ ω 𝐵 ≼ ω) | |
5 | 3, 4 | mpan 689 | 1 ⊢ (∀𝑥 ∈ ω 𝐵 ≼ ω → ∪ 𝑥 ∈ ω 𝐵 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2103 ∀wral 3063 Vcvv 3482 ∪ ciun 5019 class class class wbr 5169 ωcom 7899 ≼ cdom 8997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 ax-cc 10500 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-oi 9575 df-card 10004 df-acn 10007 |
This theorem is referenced by: ctbssinf 37321 |
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