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Mirrors > Home > MPE Home > Th. List > nnexALT | Structured version Visualization version GIF version |
Description: Alternate proof of nnex 12007, more direct, that makes use of ax-rep 5212. (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 12002 | . 2 ⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) | |
2 | rdgfun 8267 | . . 3 ⊢ Fun rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) | |
3 | omex 9429 | . . 3 ⊢ ω ∈ V | |
4 | funimaexg 6537 | . . 3 ⊢ ((Fun rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ∧ ω ∈ V) → (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) ∈ V) | |
5 | 2, 3, 4 | mp2an 688 | . 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) ∈ V |
6 | 1, 5 | eqeltri 2830 | 1 ⊢ ℕ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2101 Vcvv 3434 ↦ cmpt 5160 “ cima 5594 Fun wfun 6441 (class class class)co 7295 ωcom 7732 reccrdg 8260 1c1 10900 + caddc 10902 ℕcn 12001 |
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 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7608 ax-inf2 9427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-ov 7298 df-om 7733 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-nn 12002 |
This theorem is referenced by: zexALT 12367 qexALT 12732 reexALT 12752 |
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