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Mirrors > Home > MPE Home > Th. List > numth | Structured version Visualization version GIF version |
Description: Numeration theorem: every set can be put into one-to-one correspondence with some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Mario Carneiro, 8-Jan-2015.) |
Ref | Expression |
---|---|
numth.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
numth | ⊢ ∃𝑥 ∈ On ∃𝑓 𝑓:𝑥–1-1-onto→𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numth.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | numth2 9881 | . 2 ⊢ ∃𝑥 ∈ On 𝑥 ≈ 𝐴 |
3 | bren 8506 | . . 3 ⊢ (𝑥 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝐴) | |
4 | 3 | rexbii 3244 | . 2 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 ↔ ∃𝑥 ∈ On ∃𝑓 𝑓:𝑥–1-1-onto→𝐴) |
5 | 2, 4 | mpbi 231 | 1 ⊢ ∃𝑥 ∈ On ∃𝑓 𝑓:𝑥–1-1-onto→𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1771 ∈ wcel 2105 ∃wrex 3136 Vcvv 3492 class class class wbr 5057 Oncon0 6184 –1-1-onto→wf1o 6347 ≈ cen 8494 |
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-ac2 9873 |
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-rmo 3143 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-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-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 7103 df-wrecs 7936 df-recs 7997 df-en 8498 df-card 9356 df-ac 9530 |
This theorem is referenced by: (None) |
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