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Mirrors > Home > NFE Home > Th. List > lec0cg | GIF version |
Description: Cardinal zero is a minimal element of cardinal less than or equal. Theorem XI.2.15 of [Rosser] p. 376. (Contributed by SF, 4-Mar-2015.) |
Ref | Expression |
---|---|
lec0cg | ⊢ ((A ∈ V ∧ A ≠ ∅) → 0c ≤c A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3579 | . . . . . . 7 ⊢ ∅ ⊆ y | |
2 | 1 | jctr 526 | . . . . . 6 ⊢ (y ∈ A → (y ∈ A ∧ ∅ ⊆ y)) |
3 | 2 | eximi 1576 | . . . . 5 ⊢ (∃y y ∈ A → ∃y(y ∈ A ∧ ∅ ⊆ y)) |
4 | n0 3559 | . . . . 5 ⊢ (A ≠ ∅ ↔ ∃y y ∈ A) | |
5 | df-rex 2620 | . . . . 5 ⊢ (∃y ∈ A ∅ ⊆ y ↔ ∃y(y ∈ A ∧ ∅ ⊆ y)) | |
6 | 3, 4, 5 | 3imtr4i 257 | . . . 4 ⊢ (A ≠ ∅ → ∃y ∈ A ∅ ⊆ y) |
7 | df-0c 4377 | . . . . . 6 ⊢ 0c = {∅} | |
8 | rexeq 2808 | . . . . . 6 ⊢ (0c = {∅} → (∃x ∈ 0c ∃y ∈ A x ⊆ y ↔ ∃x ∈ {∅}∃y ∈ A x ⊆ y)) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ (∃x ∈ 0c ∃y ∈ A x ⊆ y ↔ ∃x ∈ {∅}∃y ∈ A x ⊆ y) |
10 | 0ex 4110 | . . . . . 6 ⊢ ∅ ∈ V | |
11 | sseq1 3292 | . . . . . . 7 ⊢ (x = ∅ → (x ⊆ y ↔ ∅ ⊆ y)) | |
12 | 11 | rexbidv 2635 | . . . . . 6 ⊢ (x = ∅ → (∃y ∈ A x ⊆ y ↔ ∃y ∈ A ∅ ⊆ y)) |
13 | 10, 12 | rexsn 3768 | . . . . 5 ⊢ (∃x ∈ {∅}∃y ∈ A x ⊆ y ↔ ∃y ∈ A ∅ ⊆ y) |
14 | 9, 13 | bitri 240 | . . . 4 ⊢ (∃x ∈ 0c ∃y ∈ A x ⊆ y ↔ ∃y ∈ A ∅ ⊆ y) |
15 | 6, 14 | sylibr 203 | . . 3 ⊢ (A ≠ ∅ → ∃x ∈ 0c ∃y ∈ A x ⊆ y) |
16 | 15 | adantl 452 | . 2 ⊢ ((A ∈ V ∧ A ≠ ∅) → ∃x ∈ 0c ∃y ∈ A x ⊆ y) |
17 | 0cex 4392 | . . . 4 ⊢ 0c ∈ V | |
18 | brlecg 6112 | . . . 4 ⊢ ((0c ∈ V ∧ A ∈ V) → (0c ≤c A ↔ ∃x ∈ 0c ∃y ∈ A x ⊆ y)) | |
19 | 17, 18 | mpan 651 | . . 3 ⊢ (A ∈ V → (0c ≤c A ↔ ∃x ∈ 0c ∃y ∈ A x ⊆ y)) |
20 | 19 | adantr 451 | . 2 ⊢ ((A ∈ V ∧ A ≠ ∅) → (0c ≤c A ↔ ∃x ∈ 0c ∃y ∈ A x ⊆ y)) |
21 | 16, 20 | mpbird 223 | 1 ⊢ ((A ∈ V ∧ A ≠ ∅) → 0c ≤c A) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∃wrex 2615 Vcvv 2859 ⊆ wss 3257 ∅c0 3550 {csn 3737 0cc0c 4374 class class class wbr 4639 ≤c clec 6089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-lec 6099 |
This theorem is referenced by: le0nc 6200 |
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