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Mirrors > Home > MPE Home > Th. List > r1omALT | Structured version Visualization version GIF version |
Description: Alternate proof of r1om 9515, shorter as a consequence of inar1 10046, but requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
r1omALT | ⊢ (𝑅1‘ω) ≈ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omina 9962 | . 2 ⊢ ω ∈ Inacc | |
2 | inar1 10046 | . 2 ⊢ (ω ∈ Inacc → (𝑅1‘ω) ≈ ω) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝑅1‘ω) ≈ ω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2080 class class class wbr 4964 ‘cfv 6228 ωcom 7439 ≈ cen 8357 𝑅1cr1 9040 Inacccina 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-inf2 8953 ax-ac2 9734 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-iin 4830 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-se 5406 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-isom 6237 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-2o 7957 df-oadd 7960 df-er 8142 df-map 8261 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-oi 8823 df-r1 9042 df-rank 9043 df-card 9217 df-cf 9219 df-acn 9220 df-ac 9391 df-wina 9955 df-ina 9956 |
This theorem is referenced by: (None) |
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