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Mirrors > Home > MPE Home > Th. List > nnexALT | Structured version Visualization version GIF version |
Description: Alternate proof of nnex 11722, more direct, that makes use of ax-rep 5154. (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 11717 | . 2 ⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) | |
2 | rdgfun 8081 | . . 3 ⊢ Fun rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) | |
3 | omex 9179 | . . 3 ⊢ ω ∈ V | |
4 | funimaexg 6425 | . . 3 ⊢ ((Fun rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ∧ ω ∈ V) → (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) ∈ V) | |
5 | 2, 3, 4 | mp2an 692 | . 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω) ∈ V |
6 | 1, 5 | eqeltri 2829 | 1 ⊢ ℕ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 Vcvv 3398 ↦ cmpt 5110 “ cima 5528 Fun wfun 6333 (class class class)co 7170 ωcom 7599 reccrdg 8074 1c1 10616 + caddc 10618 ℕcn 11716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 ax-inf2 9177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-nn 11717 |
This theorem is referenced by: zexALT 12082 qexALT 12446 reexALT 12466 |
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