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Mirrors > Home > MPE Home > Th. List > ex-pss | Structured version Visualization version GIF version |
Description: Example for df-pss 3623. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
ex-pss | ⊢ {1, 2} ⊊ {1, 2, 3} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ex-ss 27414 | . 2 ⊢ {1, 2} ⊆ {1, 2, 3} | |
2 | 3ex 11134 | . . . . 5 ⊢ 3 ∈ V | |
3 | 2 | tpid3 4338 | . . . 4 ⊢ 3 ∈ {1, 2, 3} |
4 | 1re 10077 | . . . . . 6 ⊢ 1 ∈ ℝ | |
5 | 1lt3 11234 | . . . . . 6 ⊢ 1 < 3 | |
6 | 4, 5 | gtneii 10187 | . . . . 5 ⊢ 3 ≠ 1 |
7 | 2re 11128 | . . . . . 6 ⊢ 2 ∈ ℝ | |
8 | 2lt3 11233 | . . . . . 6 ⊢ 2 < 3 | |
9 | 7, 8 | gtneii 10187 | . . . . 5 ⊢ 3 ≠ 2 |
10 | 6, 9 | nelpri 4234 | . . . 4 ⊢ ¬ 3 ∈ {1, 2} |
11 | nelne1 2919 | . . . 4 ⊢ ((3 ∈ {1, 2, 3} ∧ ¬ 3 ∈ {1, 2}) → {1, 2, 3} ≠ {1, 2}) | |
12 | 3, 10, 11 | mp2an 708 | . . 3 ⊢ {1, 2, 3} ≠ {1, 2} |
13 | 12 | necomi 2877 | . 2 ⊢ {1, 2} ≠ {1, 2, 3} |
14 | df-pss 3623 | . 2 ⊢ ({1, 2} ⊊ {1, 2, 3} ↔ ({1, 2} ⊆ {1, 2, 3} ∧ {1, 2} ≠ {1, 2, 3})) | |
15 | 1, 13, 14 | mpbir2an 975 | 1 ⊢ {1, 2} ⊊ {1, 2, 3} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2030 ≠ wne 2823 ⊆ wss 3607 ⊊ wpss 3608 {cpr 4212 {ctp 4214 1c1 9975 2c2 11108 3c3 11109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-2 11117 df-3 11118 |
This theorem is referenced by: (None) |
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