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Mirrors > Home > MPE Home > Th. List > nnexALT | Structured version Visualization version GIF version |
Description: Alternate proof of nnex 11445, more direct, that makes use of ax-rep 5046. (Contributed by Mario Carneiro, 3-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nnexALT | ⊢ ℕ ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nn 11439 | . 2 ⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) | |
2 | rdgfun 7855 | . . 3 ⊢ Fun rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) | |
3 | omex 8899 | . . 3 ⊢ ω ∈ V | |
4 | funimaexg 6271 | . . 3 ⊢ ((Fun rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ∧ ω ∈ V) → (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) ∈ V) | |
5 | 2, 3, 4 | mp2an 680 | . 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) ∈ V |
6 | 1, 5 | eqeltri 2857 | 1 ⊢ ℕ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2051 Vcvv 3410 ↦ cmpt 5005 “ cima 5407 Fun wfun 6180 (class class class)co 6975 ωcom 7395 reccrdg 7848 1c1 10335 + caddc 10337 ℕcn 11438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-inf2 8897 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-om 7396 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-nn 11439 |
This theorem is referenced by: zexALT 11812 qexALT 12177 reexALT 12197 |
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