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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 9850, the countable union of countable sets is countable. See iunctb 9989 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 9099 | . . 3 ⊢ ω ∈ V | |
2 | domrefg 8537 | . . 3 ⊢ (ω ∈ V → ω ≼ ω) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ω ≼ ω |
4 | iunctb 9989 | . 2 ⊢ ((ω ≼ ω ∧ ∀𝑥 ∈ ω 𝐵 ≼ ω) → ∪ 𝑥 ∈ ω 𝐵 ≼ ω) | |
5 | 3, 4 | mpan 688 | 1 ⊢ (∀𝑥 ∈ ω 𝐵 ≼ ω → ∪ 𝑥 ∈ ω 𝐵 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ∀wral 3137 Vcvv 3491 ∪ ciun 4912 class class class wbr 5059 ωcom 7573 ≼ cdom 8500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cc 9850 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 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-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-oi 8967 df-card 9361 df-acn 9364 |
This theorem is referenced by: ctbssinf 34709 |
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